Petard
A petard is a Renaissance-era bomb, basically a big firecracker: a box
or small barrel of gunpowder with a fuse attached. Those hissing
black exploding spheres that you see in Daffy Duck cartoons are
petards. Outside of cartoons, you are most likely to encounter the
petard in the phrase "hoist with his own petard", which is from
Hamlet. Rosencrantz and Guildenstern are being sent to
England with the warrant for Hamlet's death; Hamlet alters the warrant
to contain R&G's names instead of his own. "Hoist", of course,
means "raised", and Hamlet is saying that it is amusing to see someone
screw up his own petard and blow himself sky-high with it.

This morning I read in On Food in
Cooking that there's a kind of fried choux pastry called
pets de soeurs ("nuns' farts") because they're so light and
delicate. That brought to mind Le Pétomane, the world-famous
theatrical fartmaster. Then there was a link on reddit titled "Xmas
Petard (cool gif video!)" which got me thinking about petards, and it
occurred to me that "petard" was probably akin to pets, because
it makes a bang like a fart. And hey, I was right; how delightful.

Another fart-related word is "partridge", so named because its call
sounds like a fart.

Google query roundup
Now that I have a reasonably-sized body of blog posts, my blog is
starting to attract Google queries. It's really exciting when someone
visits one of my pages looking for something incredibly specific and
obscure and I can tell from their query in the referrer log that I
have unknowingly written exactly the document they were hoping to
find. That's one of the wonders of the Internet thingy.

I imagine a middle-schooler, working on her homework. The
middle-schooler is now going to have to go back to her teacher and
tell her that she was wrong, and that Franklin did not invent DST, and
a lot of other stuff that middle-school teachers usually do not want
to be bothereed with. I hope it works out well. Or perhaps the
middle-schooler will just write down "Benjamin Franklin" and leave it
at that, which would be cynical but effective.

Although you'd think that by now the middle schooler would have
figured out that questions that start with "What Pennsylvanian can
we thank for..." are about Benjamin Franklin with extremely high
probability.

The referenced page includes the title of a book that contains the
relevant essay, with a link to the bookseller. The only way the
searcher could be happier is if they found the text of the essay
itself.

Perhaps they couldn't remember the name of the Viceroy butterfly, and
my article reminded them.

Some of the queries are intriguing. I wonder what this person was
looking for?

1 spanish armada & monkey

I'd love to know the story of the Monkey and the Spanish Armada. if
there isn't one already, someone should invent one.

1 there is a cabinet with 12 drawers. each drawer is opened
only once. in each drawer are about 30 compartments, with
only 7 names.

This one was so weird that I had to do the search myself. It's a
puzzle on a page described as "Quick Riddles: Easy puzzles, riddles
and brainteasers you can solve on sight"; the question was "what is
it?" Presumably it's some sort of calendrical object, containing
pills or some other item to be dispensed daily. I looked at the
answer on the web page, which is just "the calendar". I have not seen
any calendars with drawers and compartments, so I suppose they were
meant metaphorically. I think it's a pretty crappy riddle.

Sometimes I know that the searches did not find what they were
looking for.

1 eliminate debt using linear math

I don't know what this was, but it reminds me of when I was teaching
math at the Johns Hopkins CTY program. One of my fellow instructors
told me sadly that he had a student whose uncle had invented a
brilliant secret system for making millions of dollars in the stock
market. The student had been sent to math camp to learn trigonometry
so that he would be able to execute the system for his uncle. Kids get
sent to math camp for a lot of bad reasons, but I think that one was
the winner.

1 armonica how many people can properly use it

This one is a complete miss. The armonica (or "glass harmonica") is a
kind of musical instrument. (Who can guess what Pennsylvanian we have
to thank for it?) As all ill-behaved children know, you can make a
water glass sing by rubbing its edge with a damp fingertip. The
armonica is a souped-up version of this. There is a series of glass
bowls in graduated sizes, mounted on a revolving spindle. The
operator touches the rims of the revolving bowls with his fingers;
this makes them vibrate. The smaller bowls produce higher tones. The
sound is very ethereal, not like any other instrument.

I had the good fortune to attend an armonica recital by Dean Shostak
as part of the Philadelphia Fringe Festival a few years ago.
Mr. Shostak is one of very few living armonica players. (He says
that there are seven others.) The armonica is not popular because it
is bulky, hard to manufacture, and difficult to play. The bowls must
be constructed precisely, by a skilled glassblower, to almost the
right pitch, and then carefully filed down until they are exactly
right. If you overfile one, it is junk. If a bowl goes out of tune,
it must be replaced; this requires that all the other bowls be
unmounted from the spindle. The bowls are fragile and break
easily.

The operator's hands must be perfectly clean, because the slightest
amount of lubrication prevents the operator from setting the glass
vibrating. The operator must keep his fingertips damp at all times,
continually wetting them from a convenient bowl of water. By the end
of a concert, his fingers are all pruney and have been continually
rubbed against the rotating bowls; this limits the amount of time the
instrument can be played.

Shostak's web site has some
samples that you can listen to. Unfortunately, it does not also have
any videos of him playing the instrument.

1 want did an wang invent

This one was also a miss; the poor querent found my page about
medieval Chinese type management instead.

An Wang invented the magnetic core memory that was the principal
high-speed memory for computers through the 1950s and 1960s. In this
memory technology, each bit was stored in a little ferrite doughnut,
called a "core". If the magnetic field went one way through the
doughnut, it represented a 0; the other way was a 1. Thousands of
these cores would be strung on wire grids. Each core was on one
vertical and one horizontal wire. The computer could modify the value
of the bit by sending current on the core's horizontal wire and
vertical wire simultaneously. The two currents individually were too
small to modify the other bits in the same row and column. If the bit
was actually changed, the resulting effect on the current could be
detected; this is how bits were read: You'd try to write a 1, and see
if that caused a change in the bit value. Then if it turned out to
have been a 0, you'd put it back the way it was.

The cores themselves were cheap and easy to manufacture. You mix
powdered iron with ceramic, stamp it into the desired shape in a mold,
and bake it in a kiln. Stringing cores into grids was more
expensive. and was done by hand.

As the technology improved, the cores themselves got smaller and the
grids held more and more of them. Cores from the 1950s were about a
quarter-inch in diameter; cores from the late 1960s were about
one-quarter that size. They were finally obsoleted in the 1970s by
integrated circuits.

When I was in high school in New York in the 1980s, it was still
possible to obtain ferrite cores by the pound from the
surplus-electronics stores on Canal Street. By the 1990s, the cores
were gone. You can still buy them online.

An Wang got very rich from the invention and was able to found Wang
computers. Around 1980 my mother's employer had a Wang
word-processing system. It was a marvel that took up a large space
and cost $15,000. ($35,000 in 2006 dollars.) She sometimes brought
me in on weekends so that I could play with it. Such systems, the
first word processors, were tremendously popular between 1976 and
1981. They invented the form, which, as I recall, was not
significantly different from the word processors we have today. Of
course, these systems were doomed, replaced by cheap general-purpose
machines within a few years.

The undergraduate dormitories at Harvard University are named mostly
for Harvard's presidents: Mather House, Dunster House, Eliot House,
and so on. One exception was North House. A legend says Harvard
refused an immense donation from Wang, whose successful company was
based in Cambridge, because it came with the condition that North
house be renamed after him. (Similarly, one sometimes hears it said
that the Houses are named for all the first presidents of
Harvard, except for president number 3, Leonard Hoar, who was skipped.
It's not true; numbers 2, 4, and 5 were skipped also.)

Rotten code in a ProFTPD plugin module
One of my work colleagues asked me to look at a piece of C source code
today. He was tracking down a bug in the FTP server. He thought he
had traced it to this spot, and wanted to know if I concurred and if I
agreed with his suggested change.

(You know there's something wrong when the comment says "maximal input
buffer size", but the buffer is for performing output. I have not
looked at any of the other code in this module, which is 2,800 lines
long, so I do not know if this chunk is typical.)
Mr. Colleague suggested that p=p+total_count was wrong, and
should be replaced with p=p+max_buf_size. I agreed that it
was wrong, and that his change would fix the problem, although I
suggested that p += count would be a better change.
Mr. Colleague's change, although it would no longer manifest the bug,
was still "wrong" in the sense that it would leave p pointing
to a garbage location (and incidentally invokes behavior not defined
by the C language standard) whereas my change would leave p
pointing to the end of the buffer, as one would expect.

Since this is a maintenance programming task, I recommended that we
not touch anything not directly related to fixing the bug at hand.
But I couldn't stop myself from pointing out that the code here is
remarkably badly written. Did I say "exceptionally putrid" yet? Oh,
I did.

Good. It stinks like a week-old fish.

The first thing to notice is that the expression buflen -
total_count appears four times in only nine lines of
code—five if you count the buflen > total_count
comparison. This strongly suggests that the algorithm would be more
clearly expressed in terms of whatever buflen - total_count
really is. Since buflen is the total number of characters to
be written, and total_count is the number of characters that
have been written, buflen - total_count is just the
number of characters remaining. Rather than computing the same
expression four times, we should rewrite the loop in terms of the
number of characters remaining.

Now we should notice that the two calls to gss_write are
almost exactly the same. Duplicated code like this can almost always
be eliminated, and eliminating it almost always produces a favorable
result. In this case, it's just a matter of introducing an auxiliary
variable to record the amount that should be written:

Even if we weren't here to fix a bug, we might notice something fishy:
left_to_write is being decremented by count, but
p, the buffer position, is being incremented by
total_count instead. In fact, this is exactly the bug that
was discovered by Mr. Colleague. Let's fix it:

I'm not sure I think that is an improvement. (My idea is that if we
do this, it would be better to create a p_end variable up
front, set to p+buflen, and then use p_end -
left_to_write in place of p+buflen-left_to_write. But
that adds back another variable, although it's a constant one, and the
backward logic in the calculation might be more confusing than the
thing we were replacing. Like I said, I'm not sure. What do you
think?)

Anyway, I am sure that the final code is a big improvement on the
original in every way. It has fewer bugs, both active and latent. It
has the same number of variables. It has six lines of logic instead
of eight, and they are simpler lines. I suspect that it will be a bit
more efficient, since it's doing the same thing in the same way but
without the redundant computations, although you never know what the
compiler will be able to optimize away.

The funny thing about this code is that it's performing a task that I
thought every C programmer would already have known how to do:
block-writing of a bufferfull of data. Examples of the right way to do
this are all over the place. I first saw it done in Marc
J. Rochkind's superb book Advanced Unix Programming
around 1989. (I learned from the first edition, but the link to the
right is for the much-expanded
second edition that came out in 2004.) I'm sure it must pop up all over
the Stevens books.

But the really exciting thing I've learned about code like this is
that it doesn't matter if you don't already know how to do it right,
because you can turn the wrong code into the right code, as we did
here, by noticing a few common problems, like duplicate tests and
repeated subexpressions, and applying a few simple refactorizations to
get rid of them. That's what my book will be about.

(I am also very pleased that it has taken me 37 blog entries to work
around to discussing any programming-related matters.)

A while back I was in the Penn math and physics
library browsing in the old books, and I ran across Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work by
G.H. Hardy. Srinivasa Ramanujan was an unknown amateur mathematician
in India; one day he sent Hardy some of the theorems he had been
proving. Hardy was boggled; many of Ramanujan's theorems were unlike
anything he had ever seen before. Hardy said that the formulas in the
letter must be true, because if they were not true, no one would have
had the imagination to invent them. Here's a typical example:

Hardy says that it was clear that Ramanujan was either a genius or a
confidence trickster, and that confidence tricksters of that caliber
were much rarer than geniuses, so he was prepared to give him the
benefit of the doubt.

But anyway, the main point of this note is to present the following
quotation from Hardy. He is discussing analytic number theory:

The fact remains that hardly any of Ramanujan's work in this field had
any permanent value. The analytic theory of numbers
is one of those exceptional branches of mathematics in which proof
really is everything and nothing short of absolute rigour counts.
The achievement of the mathematicians who found the Prime Number
Theorem was quite a small thing compared with that of
those who found the proof. It is not merely that in this theory (as
Littlewood's theorem shows) you can never be quite sure of the facts
without the proof, though this is important enough. The whole history
of the Prime Number Theorem, and the other big theorems
of the subject, shows that you cannot reach any real understanding of
the structure and meaning of the theory, or have any sound
instincts to guide you in further research, until you have mastered the
proofs. It is comparatively easy to make clever guesses;
indeed there are theorems like "Goldbach's Theorem", which have never
been proved and which any fool could have guessed.

(G.H. Hardy, Ramanujan.)

Some notes about this:

Notice that this implies that in most branches of
mathematics, you can get away with less than absolute rigor. I think
that Hardy is quite correct here. (This is a rather arrogant remark,
since Hardy is much more qualified than I am to be telling you what
counts as worthwhile mathematics and what it is like. But this is my
blog.) In most branches of mathematics, the difficult part is
understanding the objects you are studying. If you understand them
well enough to come up with a plausible conjecture, you are doing
well. And in some mathematical pursuits, the proof may even be
secondary. Consider, for example, linear programming problems. The
point of the theory is to come up with good numerical solutions to the
problems. If you can do that, your understanding of the mathematics
is in some sense unimportant. If you invent a good algorithm that
reliably produces good answers reasonably efficiently, proving that
the algorithm is always efficient is of rather less value. In
fact, there is such an algorithm—the "simplex
algorithm"—and it is known to have exponential time in the worst
case, a fact which is of decidedly limited practical interest.

In analytic number theory, however, two facts weigh in favor of rigor.
First, the objects you are studying are the positive integers. You
already have as much intuitive understanding of them as you are ever
going to have; you are not, through years of study and analysis, going
to come to a clearer intuition of the number 3. And second, analytic
number theory is much more inward-looking than most mathematics. The
applications to the rest of mathematics are somewhat limited, and to
the wider world even more limited. So a guessed or conjectured
theorem is unlikely to have much value; the value is in understanding
the theorem itself, and if you don't have a rigorous proof, you don't
really understand the theorem.

Hardy's example of the Goldbach conjecture is a good one. In the 18th
Century, Christian Goldbach, who was nobody in particular, conjectured
that every even number is the sum of two primes. Nobody doubts that
this is true. It's certainly true for all small even numbers, and for
large ones, you have lots and lots of primes to choose from. No
proof, however, is in view. (The primes are all about multiplication.
Proving things about their additive properties is swimming upstream.)
And nobody particularly cares whether the conjecture is true or not.
So what if every even number is the sum of two primes? But a
proof would involve startling mathematics, deep understanding of
something not even guessed at now, powerful techniques not currently
devised. The proof itself would have value, but the result
doesn't.

Fermat's theorem (the one about an +
bn = cn) is another
example of this type. Not that Fermat was in any sense a fool to have
conjectured it. But the result itself is of almost no interest.
Again, all the value is in the proof, and the techniques that were
required to carry it through.

The Prime Number Theorem that Hardy mentions is the theorem about
the average density of the prime numbers. The Greeks knew that there
were an infinite number of primes. So the next question to ask is
what fraction of integers are prime. Are the primes sparse, like the
squares? Or are they common, like multiples of 7? The answer turns
out to be somewhere in between.

Of the integers 1–10, four (2, 3, 5, 7) are prime, or 40%. Of the
integers 1–100, 25% are prime. Of the integers 1–1000, 16.8% are
prime. What's the relationship?

The relationship turns out to be amazing: Of the integers 1–n,
about 1/log(n) are prime. Here's a graph: the red line is the
fraction of the numbers 1–n that are prime; the green line is
1/log(n):

It's not hard to conjecture this, and I think it's not hard to come up
with offhand arguments why it should be so. But, as Hardy says,
proving it is another matter, and that's where the real value is,
because to prove it requires powerful understanding and sophisticated
technique, and the understanding and technique will be applicable to
other problems.

Hardy was an unusual fellow. Toward the end
of his life, he wrote an essay called A Mathematician's
Apology in which he tried to explain why he had devoted his
life for pure mathematics. I found it an extraordinarily compelling
piece of writing. I first read it in my teens, at a time when I
thought I might become a professional mathematician, and it's had a
strong influence on my life. The passage that resonates most for me
is this one:

A man who sets out to justify his existence and his activities
has to distinguish two different questions. The first is
whether the work which he does is worth doing; and the second
is why he does it, whatever its value may be, The first
question is often very difficult, and the answer very
discouraging, but most people will find the second easy enough
even then. Their answers, if they are honest, will usually
take one or another of two forms . . . the first . . . is the
only answer which we need consider seriously.

(1) 'I do what I do because it is the one and only thing I can
do at all well. . . . I agree that it might be better to be a
poet or a mathematician, but unfortunately I have no talents
for such pursuits.'

I am not suggesting that this is a defence which can be made
by most people, since most people can do nothing at all well.
But it is impregnable when it can be made without
absurdity. . . It is a tiny minority who can do anything
really well, and the number of men who can do two
things well is negligible. If a man has any genuine talent,
he should be ready to make almost any sacrifice in order to
cultivate it to the full.

And that, ultimately, is why I didn't become a mathematician.
I don't have the talent for it. I have no doubt that I could have
become a quite competent second-rate mathematician, with a secure
appointment at some second-rate college, and a series of second-rate
published papers. But as I entered my mid-twenties, it became clear
that although I wouldn't ever be a first-rate mathematician, I could
be a first-rate computer programmer and teacher of computer
programming. I don't think the world is any worse off for the lack of
my mediocre mathematical contributions. But by teaching I've been
able to give entertainment and skill to a lot of people.

When I teach classes, I sometimes come back from the mid-class break
and ask if there are any questions about anything at all. Not
infrequently, some wag in the audience asks why the sky is blue, or
what the meaning of life is. If you're going to do something as risky
as asking for unconstrained questions, you need to be ready with
answers. When people ask why the sky is blue, I reply "because it
reflects the sea." And the first time I got the question about the
meaning of life, I was glad that I had thought about this beforehand
and so had an answer ready. "Find out what your work is," I said,
"and then do it as well as you can." I am sure that this idea owes a
lot to Hardy. I wouldn't want to say that's the meaning of life for
everyone, but it seems to me to be a good answer, so if you are
looking for a meaning of life, you might try that one and see how you
like it.

(Incidentally, I'm not sure it makes sense to buy a copy of this book,
since it's really just a long essay. My copy, which is the same as
the one I've linked above, ekes it out to book length by setting it in
a very large font with very large margins, and by prepending a
fifty-page(!) introduction by C.P. Snow.)

Back on 19 January, I decided that readers might find it convenient
if, when I mentioned a book, there was a link to buy the book. I was
planning to write a lot about what books I was reading, and perhaps if
I was convincing enough about how interesting they were, people would
want their own copies.

The obvious way to do this is just to embed the HTML for the book link
directly into each entry in the appropriate place. But that is a pain
in the butt, and if you want to change the format of the book link,
there is no good way to do it. So I decided to write a Blosxom plugin
module that would translate some sort of escape code into the
appropriate HTML. The escape code would only need to contain one bit
of information about the book, say its ISBN, and then the plugin could
fetch the other information, such as the title and price, from a
database.

The initial implementation allowed me to put
<book>1558607013</book> tags into an entry, and
the plugin would translate this to the appropriate HTML. (There's an
example on the right.

) The 1558607013 was the ISBN. The
plugin would look up this key in a Berkeley DB database, where it
would find the book title and Barnes and Noble image URL. Then it
would replace the <book> element with the appropriate
HTML. I did a really bad job with this plugin and had to rewrite
it.

Since Berkeley DB only maps string keys to single string values, I had
stored the title and image URL as a single string, with a colon
character in between. That was my first dumb mistake, since book
titles frequently include colons. I ran into this right away, with
Voyages and Discoveries: Selections from Hakluyt's Principal
Navigations.

This, however, was a minor error. I had made two major errors. One
was that the <book>1558607013</book> tags were
unintelligible. There was no way to look at one and know what book
was being linked without consulting the database.

But even this wouldn't have been a disaster without the other big
mistake, which was to use Berkeley DB. Berkeley DB is a great
package. It provides fast keyed lookup even if you have millions of
records. I don't have millions of records. I will never have
millions of records. Right now, I have 15 records. In a year, I
might have 200.

The price I paid for fast access to the millions of records I don't
have is that the database is not a text file. If it were a text file,
I could look up <book>1558607013</book> by using
grep. Instead, I need a special tool to dump out the
database in text form, and pipe the output through grep. I
can't use my text editor to add a record to the database; I had to
write a special tool to do that. If I use the wrong ISBN by mistake,
I can't just correct it; I have to write a special tool to delete an
item from the database and then I have to insert the new record.

When I decided to change the field separator from colon to
\x22, I couldn't just M-x replace-string; I had to
write a special tool. If I later decided to add another field to the
database, I wouldn't be able to enter the new data by hand; I'd have
to write a special tool.

On top of all that, for my database, Berkeley DB was probably
slower than the flat text file would have been. The Berkeley
DB file was 12,288 bytes long. It has an index, which Berkeley DB
must consult first, before it can fetch the data. Loading the
Berkeley DB module takes time too. The text file is 845 bytes long
and can be read entirely into memory. Doing so requires only builtin
functions and only a single trip to the disk.

I redid the plugin module to use a flat text file with tab-separated
columns:

The columns are a nickname ("HOP" for Higher-Order Perl,
for example), the ISBN, the full title, and the image URL. The plugin
will accept either <book>1558607013</book> or
<book>HOP</book> to designate Higher-Order
Perl. I only use the nicknames now, but I let it accept ISBNs
for backward compatibility so I wouldn't have to go around changing
all the <book> elements I had already done.

Now I'm going to go off and write "just use a text file, fool!" a
hundred times.

I'm an employee of the University of Pennsylvania, and one of the best
fringe benefits of the job is that I get unrestricted access to the
library and generous borrowing privileges. A few weeks ago I was up
there, and found my way somehow into the section with the travel
books. I grabbed a bunch, one of which was the source
for my discussion of the dot product in 1580. Another was
Travels of Mirza Abu Taleb Khan, written around 1806, and
translated into English and published in English in 1814.

Travels is the account of a Persian nobleman who fell
upon hard times in India and decided to take a leave of absence and
travel to Europe. His travels lasted from 1799 through August 1803,
and when he got back to Calcutta, he wrote up an account of his
journey for popular consumption.

Wow, what a find, I thought, when I discovered it in the library. How
could such a book fail to be fascinating? But if you take that as a
real question, not as a rhetorical one, an answer comes to mind
immediately: Mirza Abu Taleb does not have very much to say!

A large portion of the book drops the names of the many people that
Mirza Abu Taleb met with, had dinner with, went riding with, went
drinking with, or attended a party at the house of. Opening the book
at random, for example, I find:

The Duke of Leinster, the first of the nobles of this kingdom
honoured me with an invitation; his house is the most superb
of any in Dublin, and contains a very numerous and valuable
collection of statues and paintings. His grace is
distinguished for the dignity of his manners, and the urbanity
of his disposition. He is blessed with several angelic
daughters.

There you see how to use sixty-two words to communicate nothing. How
fascinating it might have been to hear about the superbities of the
Duke's house. How marvelous to have seen even one of the numerous and
valuable statues. How delightful to meet one of his several angelic
daughters. How unfortunate that Abu Taleb's powers of description
have been exhausted and that we don't get to do any of those things.
"Dude, I saw the awesomest house yesterday! I can't really describe
it, but it was really really awesome!"

Here's another:

[In Paris] I also had the pleasure of again meeting my friend
Colonel Wombell, from whom I experienced so much civility in
Dublin. He was rejoiced to see me, and accompanied me to all
the public places. From Mr. and Miss Ogilvy I received the
most marked attention.

I could quote another fifty paragraphs like those, but I'll spare you.

Even when Abu Taleb has something to say, he usually doesn't say it:

I was much entertained by an exhibition of
Horsemanship, by Mr. Astley and his company. They have
an established house in London, but come over to Dublin for
four or five months in every year, to gratify the Katarah, by
displaying their skill in this science, which far surpasses
any thing I ever saw in India.

Oh boy! I can't wait to hear about the surpassing horsemanship. Did
they do tricks? How many were in the company? Was it men only, or
both men and women? Did they wear glittery costumes? What were the
horses like? Was the exhibition indoors or out? Was the crowd
pleased? Did anything go wrong?

I don't know. That's all there is about Mr. Astley and his
company.

Almost the whole book is like this. Abu Taleb is simply not a good
observer. Good writers in any language can make you feel that you
were there at the same place and the same time, seeing what they saw
and hearing what they heard. Abu Taleb doesn't understand that one
good specific story is worth a pound of vague, obscure generalities.
This defect spoils nearly every part of the book in one degree or
another:

[The Katarah] are not so intolerant as the English, neither have
they austerity and bigotry of the Scotch. In bravery and
determination, hospitality, and prodigality, freedom of speech
and open-heartedness, they surpass the English and the
Scotch, but are deficient in prudence and sound judgement:
they are nevertheless witty, and quick of comprehension.

But every once in a while you come upon an anecdote or some other
specific. I found the next passage interesting:

Thus my land lady and her children soon comprehended my broken
English; and what I could not explain by language, they
understood by signs. . . . When I was about to leave them,
and proceed on my journey, many of my friends appeared much
affected, and said: "With your little knowledge of the
language, you will suffer much distress in England; for the
people there will not give themselves any trouble to
comprehend your meaning, or to make themselves useful to you."
In fact, after I had resided for a whole year in England, and
could speak the language a hundred times better than on my
first arrival, I found much more difficulty in obtaining what
I wanted, than I did in Ireland.

Aha, so that's what he meant by "quick of comprehension". Thanks,
Mirza.

Here's another passage I liked:

In this country and all through Europe, but especially in
France and in Italy, statues of stone and marble are held in
high estimation, approaching to idolatry. Once in my
presence, in London, a figure which had lost its head, arms,
and legs, and of which, in short, nothing but the trunk
remained, was sold for 40,000 rupees (£5000). It is
really astonishing that people possessing so much knowledge
and good sense, and who reproach the nobility of Hindoostan
with wearing gold and silver ornaments like women, whould be
thus tempted by Satan to throw away their money upon useless
blocks. There is a great variety of these figures, and they
seem to have appropriate statues for every situation. . .

Oh no---he isn't going to stop there, is he? No! We're saved!

. . . thus, at the doors or gates, they have huge janitors;
in the interior they have figures of women dancing with
tambourines and other musical instruments; over the
chimney-pieces they place some of the heathen deities of
Greece; in the burying grounds they have the statues of the
deceased; and in the gardens they put up devils, tigers, or
wolves in pursuit of a fox, in hopes that animals, on
beholding these figures will be frightened, and not come into
the garden.

If more of the book were like that, it would be a treasure. But you
have to wait a long time between such paragraphs.

There are plenty of good travel books in the
world. Kon-Tiki, for example. In Kon-Tiki,
Thor Heyerdahl takes you across the Pacific Ocean on a balsa wood
raft. Every detail is there: how and why they built the raft, and the
troubles they went to to get the balsa, and to build it, and to launch
it. How it was steered, and where they kept the food and water. What
happened to the logs as they got gradually more waterlogged and the
incessant rubbing of the ropes ropes wore them away. What they ate,
and drank, and how they cooked and slept and shat. What happened in
storms and calm. The fish that came to visit, and how every morning
the first duty of the day's cook was to fry up the flying fish that
had landed on the roof of the cabin in the night. Every page has some
fascinating detail that you would not have been able to invent
yourself, and that's what makes it worth reading, because what's the
point of reading a book that you could have invented yourself?

Another similarly good travel book is Sir Richard Francis Burton's
1853 account of his pilgimage to Mecca. Infidels were not allowed in the
holy city of Mecca. Burton disguised himself as an Afghan and snuck
in. I expect I'll have something to say about this book in a future
article.

The octopus and the creation of the cosmosIn
an earlier post, I mentioned the lucky finds you sometimes make
when you're wandering at random in a library. Here's another such.
In 2001 I was in Boston with my wife, who was attending the United
States Figure Skating Championships. Instead of attending the Junior
Dance Compulsories, I went to the Boston Public Library, where I
serendipitously unearthed the following treasure:

Although we have the source of all things from chaos, it is a
chaos which is simply the wreck and ruin of an earlier
world....The drama of creation, according to The Hawaiian
account, is divided into a series of stages, and in the very
first of these life springs from the shadowy abyss and dark
night...At first the lowly zoophytes and corals come into
being, and these are followed by worms and shellfish, each
type being declared to conquer and destroy its predecessor, a
struggle for existence in which the strongest survive....As
type follows type, the accumulating slime of their decay
raises land above the waters, in which, as spectator of all,
swims the octopus, the lone survivor of an earlier world.

More irrational numbers
Gaal Yahas has written in with a delightfully simple proof that a
particular number is irrational. Let x =
log2 3; that is, such that 2x = 3. If
x is rational, then we have 2a/b = 3
and 2a = 3b, where a and
b are integers. But the left side is even and the right side
is odd, so there are no such integers, and x must be
irrational.

As long as I am on the subject, undergraduates are sometimes asked
whether there are irrational numbers a and b such
that ab is rational. It's easy to prove that
there are. First, consider a = b = √2. If
√2√2 is rational, then we are done. Otherwise,
take a = √2√2 and b = √2.
Both are irrational, but ab = 2.

This is also a standard example of a non-constructive proof: it
demonstrates conclusively that the numbers in question exist, but it
does not tell you which of the two constructed pairs is actually the
one that is wanted. Pinning down the real answer is tricky. The Gelfond-Schneider
theorem establishes that it is in fact the second pair, as one
would expect.

The square root of 2 is irrational
I heard some story that the Pythagoreans tried to cover this up by
drowning the guy who discovered it, but I don't know if it's true and
probably nobody else does either.

The usual proof goes like this. Suppose that √2 is rational;
then there are integers a and b with a / b
= √2, where a / b is in lowest terms. Then
a2 / b2 = 2, and
a2 = 2b2. Since the right-hand
side is even, so too must the left-hand side be, and since
a2 is even, a must also be even. Then
a = 2k for some integer k, and we have
4k2 = 2b2, and so
2k2 = b2. But then since the
left-hand side is even, so too must the right-hand side be, and since
b2 is even, b must also be even. But since
a and b are both even, a / b was not in
lowest terms, a contradiction. So no such a and b can
exist, and √2 is irrational.

There are some subtle points that are glossed over here, but that's
OK; the proof is correct.

A number of years ago, a different proof occurred to me. It goes like
this:

Suppose that √2 is rational; then there are integers a
and b with a / b = √2, where a /
b is in lowest terms. Since a and b have no
common factors, nor do a2 and b2,
and a2 / b2 = 2 is also in lowest
terms. Since the representation of rational numbers by fractions in
lowest terms is unique, and a2 /
b2 = 2/1, we have a2 = 2. But
there is no such integer a, a contradiction. So no such
a and b can exist, and √2 is irrational.

This also glosses over some subtle points, but it also seems to be
correct.

I've been pondering this off and on for several years now, and it
seems to me that it seems simpler in some ways and more complex in
others. These are all hidden in the subtle points I alluded to.

For example, consider fact that both proofs should go through just as
well for 3 as for 2. They do. And both should fail for 4, since
√4 is rational. Where do these failures occur?
The first proof concludes that since a2 is even,
a must be also. This is simple. And this is the step that
fails if you replace 2 with 4: the corresponding deduction is that
since a2 is a multiple of 4, a must be also.
This is false. Fine.

You would also like the proof to go through successfully for 12,
because √12 is irrational. But instead it fails, because the
crucial step is that since a2 is divisible by 12,
a must be also—and this step is false.

You can fix this, but you have to get tricky. To make it go through
for 12, you have to say that a2 is divisible by
3, and so a must be also. To do it in general for
√n requires some fussing.

The second proof, however, works whenever it should and fails whenever
it shouldn't. The failure for √4 is in the final step, and it
is totally transparent: "we have a2 = 4," it says,
"but there is no such integer....oops, yes there is." And, unlike the
first proof, it works just fine for 12, with no required fussery: "we
have a2 = 12. But there is no such integer, a
contradiction."

The second proof depends on the (unproved) fact that lowest-term
fractions are unique. This is actually a very strong theorem. It is
true in the integers, but not in general domains. (More about this in
the future, probably.) Is this a defect? I'm not sure. On the one
hand, one could be seen as pulling the wool over the readers' eyes, or
using a heavy theorem to prove a light one. On the other hand, this
is a very interesting connection, and raises the question of whether
the corresponding theorems are true in general domains. The first
proof also does some wool-pulling, and it's rather more
complicated-looking than the second. And whereas the first one
appears simple, and is actually more complex than it seems, the point
of complexity in the second proof is right out in the open, inviting
question.

The really interesting thing here is that you always see the
first proof quoted, never the second. When I first discovered the
second proof I pulled a few books off the shelf at random to see how
the proof went; it was invariably the first one. For a while I
wondered if perhaps the second proof had some subtle mistake I was
missing, but I'm pretty sure it doesn't.

I looked it up in the dictionary, and it turns out it's simple.
"Farther" means "more far". "Further" means "more forward".

"Further" does often connote "farther", because something that is
further out is usually farther away, and so in many cases the two are
interchangeable. For example, "Hitherto shalt thou come, but no
further" (Job 38:11.)

Google finds 3.2 million citations for "further back", and 9.5 million
for "further behind", so common usage is strongly in favor of this.
But a quick check of the OED does not reveal much historical confusion
between these two. Of the citations there, I can only find one that
rings my alarm bell. ("1821 J. BAILLIE Metr. Leg., Wallace lvi, In the
further rear.")

Morphogenetic puzzles
In a recent
post, I briefly discussed puzzling issues of morphogenesis: when a
caterpillar pupates, how do its cells know how to reorganize into a
butterfly? When the blastocyst grows inside a mammal, how do its
cells know what shape to take? I said it was all a big mystery.

A reader, who goes by the name of Omar, wrote to remind me of the
"Hox" (short for "homeobox") genes discussed by Richard Dawkins in
The Ancestor's Tale. (No "buy this" link; I only do that
for books I've actually read and recommend.) These genes are
certainly part of the story, just not the part I was wondering
about.

The Hox genes seem to be the master controls for notifying developing
cells of their body locations. The proteins they manufacture bind
with DNA and enable or disable other genes, which in turn manufacture
proteins that enable still other genes, and so on. A mutation to the
Hox genes, therefore, results in a major change to the animal's body
plan. Inserting an additional copy of a Hox gene into an invertebrate
can cause its offspring to have duplicated body segements; transposing
the order of the genes can mix up the segments. One such mutation,
occurring in fruit flies, is called antennapedia, and causes the flies' antennae to be
replaced by fully-formed legs!

So it's clear that these genes play an important part in the overall
body layout.

But the question I'm most interested in right now is how the
small details are implemented. That's why I specifically
brought up the example of a ring finger.

Or consider that part of the ring finger turns into a fingernail bed
and the rest doesn't. The nail bed is distally located, but the
most distal part of the finger nevertheless decides not
to be a nail bed. And the ventral part of the finger at the same
distance also decides not to be a nail bed.

Meanwhile, the ear is growing into a very complicated but specific
shape with a helix and an antihelix and a tragus and an antitragus.
How does that happen? How do the growing parts communicate between
each other so as to produce that exact shape? (Sometimes, of course,
they get confused; look up accessory
tragus for example.)

In computer science there are a series of related problems called
"firing squad problems". In the basic problem, you have a line of
soldiers. You can communicate with the guy at one end, and other than
that each soldier can only communicate with the two standing next to
him. The idea is to give the soldiers a protocol that allows them to
synchronize so that they all fire their guns simultaneously.

It seems to me that the embryonic cells have a much more difficult
problem of the same type. Now you need the soldiers to get into an
extremely elaborate formation, even though each soldier can only see
and talk to the soldiers next to him.

Omar suggested that the Hox genes contain the answer to how the fetal
cells "know" whether to be a finger and not a kneecap. But I think
that's the wrong way to look at the problem, and one that glosses over
the part I find so interesting. No cell "becomes a finger". There is
no such thing as a "finger cell". Some cells turn into hair follicles
and some turn into bone and some turn into nail bed and some turn into
nerves and some turn into oil glands and some turn into fat, and yet
you somehow end up with all the cells in the right places turning into
the right things so that you have a finger! And the finger has hair
on the first knuckle but not the second. How do the cells know which
knuckle they are part of? At the end of the finger, the oil glands
are in the grooves and not on the ridges. How do the cells know
whether they will be at the ridges or the grooves? And the fat pad is
on the underside of the distal knuckle and not all spread around. How
do the cells know that they are in the middle of the ventral surface
of the distal knuckle, but not too close to the surface?

Somehow the fat pad arises in just the right place, and decides to
stop growing when it gets big enough. The hair cells arise only on
the dorsal side and the oil glands only on the ventral side.

How do they know all these things? How does the cell decide that it's
in the right place to differentiate into an oil gland cell? How does
the skin decide to grow in that funny pattern of ridges and grooves?
And having decided that, how do the skin cells know whether they're
positioned at the appropriate place for a ridge or a groove? Is there
a master control that tells all the cells everything at once? I bet
not; I imagine that the cells conduct chemical arguments with their
neighbors about who will do which job.

One example of this kind of communication is phyllotaxis, the way
plants decide how to distribute their leaves around the stem. Under
certain simple assumptions, there is an optimal way to do this: you
want to go around the stem, putting each leaf about 360°/φ
farther than the previous one, where φ is ½(1+√5).
(More about this in some future post.) And in fact many plants do
grow in just this pattern. How does the plant do such an elaborate
calculation? It turns out to be simple: Suppose leafing is controlled
by the buildup of some chemical, and a leaf comes out when the
chemical concentration is high. But when a leaf comes out, it also
depletes the concentration of the chemical in its vicinity, so that
the next leaf is more likely to come out somewhere else. Then the
plant does in fact get leaves with very close to optimal placement.
Each leaf, when it comes out, warns the nearby cells not to turn into
a leaf themselves---not until the rest of the stem is full, anyway. I
imagine that the shape of the ear is constructed through a more
complicated control system of the same sort.

Red Flags world tour: New York City
My wife came up with a brilliant plan to help me make regular progress
on my current book.
The idea of the book is that I show how to take typical programs and
repair and refurbish them. The result usually has between one-third
and one-half less code, is usually a little faster, and sometimes has
fewer bugs. Lorrie's idea was that I should schedule a series of talks for
Perl Mongers groups. Before each talk, I would solicit groups to
send me code to review; then I'd write up and deliver the talk, and
then afterward I could turn the talk notes into a book chapter. The
talks provide built-in, inflexible deadlines, and I love giving talks,
so the plan will help keep me happy while writing the book.

'It makes no odds whether a man has a thousand pound, or nothing,
there. Particular in New York, I'm told, where Ned landed.'

'New York, was it?' asked Martin, thoughtfully.

'Yes,' said Bill. 'New York. I know that, because he sent word home
that it brought Old York to his mind, quite wivid, in consequence of
being so exactly unlike it in every respect.'

(Charles Dickens, Martin
Chuzzlewit, about which more in some future entry, perhaps.)

The New Yorkers gave me a wonderful welcome, and generously paid my
expenses afterward. The only major hitch was that I accidentally
wrote my talk about a submission that had come from London. Oops! I
must be more careful in the future.

Each time I look at a new program it teaches me something new. Some
people, perhaps, seem to be able to reason from general principles to
specifics: if you tell them that common code in both branches of a
conditional can be factored out, they will immediately see what you
mean. Or so they would have you believe; I have my doubts. Anyway,
whether they are telling the truth or not, I have almost none of that
ability myself. I frequently tell people that I have very little
capacity for abstract thought. They sometimes think I'm joking, but
I'm not. What I mean is that I can't identify, remember, or
understand general principles except as generalizations of specific
examples. Whenever I want to study some problem, my approach is
always to select a few typical-seeming examples and study them
minutely to try to understand what they might have in common. Some
people seem to be able to go from abstract properties to conclusions;
I can only go from examples.

So my approach to understanding how to improve programs is to collect
a bunch of programs, repair them, take notes, and see what sorts of
repairs come up frequently, what techniques seem to apply to multiple
programs, what techniques work on one program and fail on another, and
why, and so on. Probably someone smarter than me would come up with a
brilliant general theory about what makes bad programs bad, but that's
not how my brain works. My brain is good at coming up with a body of
technique. It's a limitation, but it's not all bad.

The goal of generalization had become so fashionable that a generation
of mathematicians had become unable to relish beauty in the
particular, to enjoy the challenge of solving quantitative problems,
or to appreciate the value of technique.

So anyway, here's something I learned from this program. I have this
idea now that you should generally avoid the Perl . (string
concatenation) operator, because there's almost always a better
alternative. The typical use of the . operator looks like
this:

$html = "<a href='".$url."'>".$hot_text."</a>";

It's hard to see here what is code and what is data. You
pretty much have to run the Perl lexer algorithm in your head. But
Perl has another notation for concatenating strings: "$a$b"
concatenates strings $a and $b. If you use this
interpolation notation to rewrite the example above, it gets much
easier to read:

$html = "<a href='$url'>$hot_text</a>";

So when I do these classes, I always suggest that whenever you're
going to use the . operator, you try writing it as an
interpolation too and see which you like better.

This frequently brings on a question about what to do in cases like
this:

$tmpfilealrt = "alert_$daynum" . "_$day" . "_$mon.log" ;

Here you can't eliminate the . operators in this way, because
you would get:

$tmpfilealrt = "alert_$daynum_$day_$mon.log" ;

This fails because it wants to interpolate $daynum_ and
$day_, rather than $daynum and $day. Perl
has an escape hatch for this situation:

$tmpfilealrt = "alert_${daynum}_${day}_$mon.log" ;

But it's not clear to me that that is an improvement on the version
that used the . operator. The punctuation is only slightly
reduced, and you've used an obscure notation that a lot of people
won't recognize and that is visually similar to, but entirely
unconnected with, hash notation.

Anyway, when this question would come up, I'd discuss it, and say that
yeah, in that case it didn't seem to me that the . operator
was inferior to the alternatives. But since my review of the program
I talked about in New York on Monday, I know a better
alternative. The author of that program wrote it like this:

In an
earlier post I remarked that "The liver of arctic animals . . .
has a toxically high concentration of vitamin D". Dennis Taylor has
pointed out that this is mistaken; I meant to say "vitamin A".
Thanks, Dennis.

B and C vitamins are not toxic in large doses; they are water-soluble
so that excess quantities are easily excreted. Vitamins A and D are
not water-soluble, so excess quantities are harder to get rid of.
Apparently, though, the liver is capable of storing very large
quantities of vitamin D, so that vitamin D poisoning is extremely
rare.

The only cases of vitamin A poisoning I've heard of concerned either
people who ate the livers of polar bears, walruses, sled dogs, or
other arctic animals, or else health food nuts who consumed enormous
quantities of pure vitamin A in a misguided effort to prove how
healthy it is. In On Food and Cooking, Harold McGee writes:

In the space of 10 days in February of 1974, an English health food
enthusiast named Basil Brown took about 10,000 times the recommended
requirement of vitamin A, and drank about 10 gallons of carrot juice,
whose pigment is a precursor of vitamin A. At the end of those ten
days, he was dead of severe liver damage. His skin was bright
yellow.

(First edition, p. 536.)

There was a period in my life in which I was eating very large
quantities of carrots. (Not for any policy reason; just because I
like carrots.) I started to worry that I might hurt myself, so I did
a little research. The carrots themselves don't contain vitamin A;
they contain beta-carotene, which the body converts internally to
vitamin A. The beta-carotene itself is harmless, and excess is easily
eliminated. So eat all the carrots you want! You might turn orange,
but it probably won't kill you.

Butterflies
Yesterday I visited the American Museum of Natural History in New York
City, for the first time in many years. They have a special exhibit
of butterflies. They get pupae shipped in from farms, and pin the
pupae to wooden racks; when the adults emerge, they get to flutter
around in a heated room that is furnished with plants, ponds of
nectar, and cut fruit.

The really interesting thing I learned was that chrysalises are not
featureless lumps. You can see something of the shape of the animal
in them. (See, for example, this
Wikipedia illustration.) The caterpillar has an exoskeleton,
which it molts several times as it grows. When time comes to pupate,
the chrysalis is in fact the final exoskeleton, part of the animal
itself. This is in contrast to a cocoon, which is different. A
cocoon is a case made of silk or leaves that is not part of the
animal; the animal builds it and lives inside. When you think of a
featureless round lump, you're thinking of a cocoon.

Until recently, I had the idea that the larva's legs get longer, wings
sprout, and so forth, but it's not like that at all. Instead, inside
the chrysalis, almost the entire animal breaks down into a liquid!
The metamorphosis then reorganizes this soup into an adult. I asked
the explainer at the Museum if the individual cells retained their
identities, or if they were broken down into component chemicals. She
didn't know, unfortunately. I hope to find this out in coming weeks.

How does the animal reorganize itself during metamorphosis? How does
its body know what new shape to grow into? It's all a big mystery.
It's nice that we still have big mysteries. Not all mysteries have
survived the scientific revolution. What makes the rain fall and the
lightning strike? Solved problems. What happens to the food we eat,
and why do we breathe? Well-understood. How does the butterfly
reorganize itself from caterpillar soup? It's a big puzzle.

A related puzzle is how a single cell turns into a human baby during
gestation. For a while, the thing doubles, then doubles again, and
again, becoming roughly spherical, as you'd expect. But then stuff
starts to happen: it dimples, and folds over; three layers form, a
miracle occurs, and eventually you get a small but perfectly-formed
human being. How do the cells in the fingers decide to turn into
fingers? How does the cells in the fourth finger know they're one
finger from one side of the hand and three fingers from the other
side? Maybe the formation of the adult insect inside the chrysalis
uses a similar mechanism. Or maybe it's completely different. Both
possibilities are mind-boggling.

This is nowhere near being the biggest pending mystery; I think we at
least have some idea of where to start looking for the answer.
Contrast this with the question of how it is we are conscious, where
nobody even has a good idea of what the question is.

Other caterpillar news: chrysalides are so named because they often
have a bright golden sheen, or golden features. (Greek "khrusos" is
"gold".) The
Wikipedia picture of this is excellent too. The "gold" is a
yellow pigmented area covered with a shiny coating. The explainer
said that some people speculate that it helps break up the outlines of
the pupa and camouflage it.

I asked if the chrysalis of the viceroy butterfly, which, as an adult,
resembles the poisonous monarch butterfly, also resembled the
monarch's chrysalis. The answer: no, they look completely different.
Isn't that interesting? You'd think that the pupa would get at least
as much benefit from mimicry as the adult. One possible explanation
why not: most pupae don't make it to adulthood anyway, so the marginal
benefit to the species from mimicry in the pupal stage is small
compared with the benefit in the adult stage. Another: the pupa's
main defense, which is not available to the adult, is to be difficult
to see; beyond that it doesn't matter much what happens if it
is seen. Which is correct? I don't know.

For a long time folks thought that the monarch was poisonous and the
viceroy was not, and that the viceroy's monarch-like coloring tricked
predators into avoiding it unnecessarily. It's now believed that
both speciies are poisonous and bad-tasting, and that their
similar coloring therefore protects both species. A predator who eats
one will avoid both in the future. The former kind of mimicry is
called Batesian; the latter, Müllerian.

The monarch butterfly does not manufacture its toxic and bad-tasting
chemicals itself. It is poisonous because it ingests poisonous
chemicals in its food, which I think is milkweed plants. Plant
chemistry is very weird. Think of all the poisonous foods you've ever
heard of. Very few of them are animals. (The only poisonous meat I
can think of offhand is the liver of arctic animals, which has a
toxically high concentration of vitamin D.) If you're stuck on a
desert island, you're a lot safer eating strange animals than you are
eating strange berries.

The essential feature of DST is that there is an official change to
the civil calendar to move back all the real times by one hour.
Events that were scheduled to occur at noon now occur at 11 AM,
because all the clocks say noon when it's really 11 AM.

The proposal by Franklin that's cited as evidence that he invented DST
doesn't propose any such thing. It's a letter to the editors of
The Journal of Paris, originally sent in 1784. There
are two things you should know about this letter: First, it's
obviously a joke. And second, what it actually proposes is
just that people should get up earlier!

I went home, and to bed, three or four hours after
midnight. . . . An accidental sudden noise waked me about six
in the morning, when I was surprised to find my room filled
with light. . . I got up and looked out to see what might be
the occasion of it, when I saw the sun just rising above the
horizon, from whence he poured his rays plentifully into my
chamber. . .

. . . still thinking it something extraordinary that the sun
should rise so early, I looked into the almanac, where I found
it to be the hour given for his rising on that day. . . . Your
readers, who with me have never seen any signs of sunshine
before noon, and seldom regard the astronomical part of the
almanac, will be as much astonished as I was, when they hear
of his rising so early; and especially when I assure them,
that he gives light as soon as he rises. I am convinced
of this. I am certain of my fact. One cannot be more certain
of any fact. I saw it with my own eyes. And, having repeated
this observation the three following mornings, I found always
precisely the same result.

I considered that, if I had not been awakened so early in the
morning, I should have slept six hours longer by the light of
the sun, and in exchange have lived six hours the following
night by candle-light; and, the latter being a much more
expensive light than the former, my love of economy induced me
to muster up what little arithmetic I was master of, and to
make some calculations. . .

Franklin then follows with a calculation of the number of candles that
would be saved if everyone in Paris got up at six in the morning
instead of at noon, and how much money would be saved thereby. He
then proposes four measures to encourage this: that windows be taxed
if they have shutters; that "guards be placed in the shops of the wax
and tallow chandlers, and no family be permitted to be supplied with
more than one pound of candles per week", that travelling by coach
after sundown be forbidden, and that church bells be rung and cannon
fired in the street every day at dawn.

Franklin finishes by offering his brilliant insight to the world free
of charge or reward:

I expect only to have the honour of it. And yet I know there
are little, envious minds, who will, as usual, deny me this
and say, that my invention was known to the ancients, and
perhaps they may bring passages out of the old books in proof
of it. I will not dispute with these people, that the ancients
knew not the sun would rise at certain hours; they possibly
had, as we have, almanacs that predicted it; but it does not
follow thence, that they knew he gave light as soon as he
rose. This is what I claim as my discovery.

OK, I'm not done yet. I think the story of how I happened to find
this out might be instructive.

I used to live at 9th and Pine streets, across from Pennsylvania
Hospital. (It's the oldest hospital in the U.S.) Sometimes I would
get tired of working at home and would go across the street to the
hospital to read or think. Hospitals in general are good for that:
they are well-equipped with lounges, waiting rooms, comfortable
chairs, sofas, coffee carts, cafeterias, and bathrooms. They are open
around the clock. The staff do not check at the door to make sure
that you actually have business there. Most of the people who work in
the hospital are too busy to notice if you have been hanging around
for hours on end, and if they do notice they will not think it is
unusual; people do that all the time. A hospital is a great place to
work unmolested.

Pennsylvania Hospital
is an unusually pleasant hospital. The original building is still
standing, and you can go see the cornerstone that was laid in 1755 by
Franklin himself. It has a beautful flower garden, with azaleas and
wisteria, and a medicinal herb garden. Inside, the building is
decorated with exhibits of art and urban archaeology, including a fire
engine that the hospital acquired in 1780, and a massive painting of
Christ healing the sick, originally painted by Benjamin West so that
the hospital could raise funds by charging people a fee to come look
at it. You can visit the 19th-century surgical amphitheatre, with its
observation gallery. Even the food in the cafeteria is way above
average. (I realize that that is not saying much, since it is, after
all, a hospital cafeteria. But it was sufficiently palatable to
induce me to eat lunch there from time to time.)

Having found so many reasons to like Pennsylvania Hospital, I went to
visit their web site to see what else I could find out. I discovered
that the hospital's clinical library, adjacent to the surgical
amphitheatre, was open to the public. So I went to visit a few times
and browsed the stacks.

Mostly, as you would expect, they had a lot of medical texts. But
on one of these visits I happened to notice a
copy of Ingenious Dr. Franklin: Selected Scientific Letters of
Benjamin Franklin on the shelf. This caught my interest, so I
sat down with it. It contained all sorts of good stuff, including
Franklin's letter on "Daylight Saving". Here is the table of contents:

Preface
The Ingenious Dr. Franklin
Daylight Saving
Treatment for Gout
Cold Air Bath
Electrical Treatment for Paralysis
Lead Poisoning
Rules of Health and Long Life
The Art of Procuring Pleasant Dreams
Learning to Swim
On Swimming
Choosing Eye-Glasses
Bifocals
Lightning Rods
Advantage of Pointed Conductors
Pennsylvanian Fireplaces
Slaughtering by Electricity
Canal Transportation
Indian Corn
The Armonica
First Hydrogen Balloon
A Hot-Air Balloon
First Aerial Voyage by Man
Second Aerial Voyage by Man
A Prophecy on Aerial Navigation
Magic Squares
Early Electrical Experiments
Electrical Experiments
The Kite
The Course and Effect of Lightning
Character of Clouds
Musical Sounds
Locating the Gulf Stream
Charting the Gulf Stream
Depth of Water and Speed of Boats
Distillation of Salt Water
Behavior of Oil on Water
Earliest Account of Marsh Gas
Smallpox and Cancer
Restoration of Life by Sun Rays
Cause of Colds
Definition of a Cold
Heat and Cold
Cold by Evaporation
On Springs
Tides and Rivers
Direction of Rivers
Salt and Salt Water
Origin of Northeast Storms
Effect of Oil on Water
Spouts and Whirlwinds
Sun Spots
Conductors and Non-Conductors
Queries on Electricity
Magnetism and the Theory of the Earth
Nature of Lightning
Sound
Prehistoric Animals of the Ohio
Toads Found in Stone
Checklist of Letters and Papers
List of Correspondents
List of a Few Additional Letters

I'm sure that anyone who bothers to read my blog would find at least
some of those items appealing. I certainly did.

Anyway, the moral of the story, as I see it, is: If you make your way
into strange libraries and browse through the stacks, sometimes you
find some good stuff, so go do that once in a while.

In 1920 Hugh Lofting wrote and illustrated The Story of Doctor
Dolittle, an account of a small-town English doctor around 1840
who learns to speak the languages of animals and becomes the most
successful veterinarian the world has ever seen. The book was a
tremendous success, and spawned thirteen sequels, two posthumously.
The 1922 sequel, The Voyages of Doctor Dolittle, won the
prestigious Newbery award.
The books have been reprinted many times, and the first two are now in
the public domain in the USA, barring any further meddling by Congress
with the copyright statute. The Voyages of Doctor
Dolittle was one of my favorite books as a child, and I know it
by heart. I returned the original 1922 copy that I had to my
grandmother shortly before she died, and replaced it with a 1988
reprinting, the "Dell Centenary Edition". On reading the new
copy, I discovered that some changes had been made to the text—I had
heard that a recent edition of the books had attempted to remove
racist references from them, and I discovered that my new 1988 copy
was indeed this edition.

The 1988 reprinting contains an afterword by Christopher Lofting, the
son of Hugh Lofting, and explains why the changes were made:

When it was decided to reissue the Doctor Dolittle books, we were
faced with a challenging opportunity and decision. In some of the
books there were certain incidents depicted that, in light of today's
sensitivities, were considered by some to be disrespectful to ethnic
minorities and, therefore, perhaps inappropriate for today's young
reader. In these centenary editions, this issue is addressed.

. . . After much soul-searching the consensus was that changes should
be made. The deciding factor was the strong belief that the author
himself would have immediately approved of making the alterations.
Hugh Lofting would have been appalled at the suggestion that any part
of his work could give offense and would have been the first to have
made the changes himself. In any case, the alterations are minor
enough not to interfere with the style and spirit of the original.

This note will summarize some of the changes to The Voyages of
Doctor Dolittle. I have not examined the text exhaustively. I
worked from memory, reading the Centenary Edition, and when I thought
I noticed a change, I crosschecked the text against the Project
Gutenberg version of the original text. So this does not purport to
be a complete listing of all the changes that were made. But I do
think it is comprehensive enough to give a sense of what was changed.

Many of the changes concern Prince Bumpo, a character who first
appeared in The Story of Doctor Dolittle. Bumpo is a
black African prince, who, at the beginning of Voyages,
is in England, attending school at Oxford.
Bumpo is a highly sympathetic character, but also a comic one. In
Voyages his speech is sprinkled with inappropriate
"Oxford" words: he refers to "the college quadrilateral", and later
says "I feel I am about to weep from sediment", for example. Studying
algebra makes his head hurt, but he says "I think Cicero's fine—so
simultaneous. By the way, they tell me his son is rowing for our
college next year—charming fellow." None of this humor at Bumpo's
expense has been removed from the Centenary Edition.

Bumpo's first appearance in the book, however, has been substantially
cut:

The Doctor had no sooner gone below to stow away his
note-books than another visitor appeared upon the
gang-plank. This was a most extraordinary-looking black
man. The only other negroes I had seen had been in circuses,
where they wore feathers and bone necklaces and things like
that. But this one was dressed in a fashionable frock coat
with an enormous bright red cravat. On his head was a straw
hat with a gay band; and over this he held a large green
umbrella. He was very smart in every respect except his
feet. He wore no shoes or socks.

In the revised edition, this is abridged to:

The Doctor had no sooner gone below to stow away his
note-books than another visitor appeared upon the gang-plank.
This was a black man, very fashionably dressed.
(p. 128)

I think it's interesting that they excised the part about Bumpo being
barefooted, because the explanation of his now unmentioned
barefootedness still appears on the following page. (The shoes hurt
his feet, and he threw them over the wall of "the college
quadrilateral" earlier that morning.) Bumpo's feet make another
appearance later on:

I very soon grew to be quite fond of our funny black friend
Bumpo, with his grand way of speaking and his enormous feet
which some one was always stepping on or falling over.

The only change to this in the revised version is the omission of the
word 'black'. (p.139)

This is typical. Most of the changes are excisions of rather ordinary
references to the skin color of the characters. For example, the
original:

It is quite possible we shall be the first white men to land
there. But I daresay we shall have some difficulty in finding
it first."

The bowdlerized version omits 'white men'. (p.120.)

Another typical cut:

"Great Red-Skin," he said in the fierce screams and short grunts
that the big birds use, "never have I been so glad in all my life
as I am to-day to find you still alive."

In a flash Long Arrow's stony face lit up with a smile of
understanding; and back came the answer in eagle-tongue.

"Mighty White Man, I owe my life to you. For the remainder of my
days I am your servant to command."

(Long Arrow has been buried alive for several months in a cave.) The
revised edition replaces "Great Red-Skin" with "Great Long Arrow", and
"Mighty White Man" with "Mighty Friend". (p.223)

Another, larger change of this type, where apparently value-neutral
references to skin color have been excised, is in the poem "The Song
of the Terrible Three" at the end of part V, chapter 5. The complete
poem is:

THE SONG OF THE TERRIBLE THREE

Oh hear ye the Song of the Terrible Three
And the fight that they fought by the edge of the sea.
Down from the mountains, the rocks and the crags,
Swarming like wasps, came the Bag-jagderags.

Surrounding our village, our walls they broke down.
Oh, sad was the plight of our men and our town!
But Heaven determined our land to set free
And sent us the help of the Terrible Three.

One was a Black—he was dark as the night;
One was a Red-skin, a mountain of height;
But the chief was a White Man, round like a bee;
And all in a row stood the Terrible Three.

Shoulder to shoulder, they hammered and hit.
Like demons of fury they kicked and they bit.
Like a wall of destruction they stood in a row,
Flattening enemies, six at a blow.

Oh, strong was the Red-skin fierce was the Black.
Bag-jagderags trembled and tried to turn back.
But 'twas of the White Man they shouted, "Beware!
He throws men in handfuls, straight up in the air!"

Long shall they frighten bad children at night
With tales of the Red and the Black and the White.
And long shall we sing of the Terrible Three
And the fight that they fought by the edge of the sea.

The ten lines in boldface have been excised in the revised
version. Also in this vicinity, the phrase "the strength and weight of
those three men of different lands and colors" has been changed to
omit "and colors". (pp. 242-243)

Here's an interesting change:

Long Arrow said they were apologizing and trying to tell the
Doctor how sorry they were that they had seemed unfriendly to
him at the beach. They had never seen a white man before and
had really been afraid of him—especially when they saw him
conversing with the porpoises. They had thought he was the
Devil, they said.

In some cases the changes seem completely bizarre. When I first heard
that the books had been purged of racism I immediately thought of this
passage, in which the protagonists discover that a sailor has stowed
away on their boat and eaten all their salt beef (p. 142):

"I don't know what the mischief we're going to do now," I
heard her whisper to Bumpo. "We've no money to buy any more;
and that salt beef was the most important part of the stores."

"Would it not be good political economy," Bumpo whispered
back, "if we salted the able seaman and ate him instead? I
should judge that he would weigh more than a hundred and
twenty pounds."

"How often must I tell you that we are not in Jolliginki,"
snapped Polynesia. "Those things are not done on white men's
ships—Still," she murmured after a moment's thought, "it's an
awfully bright idea. I don't suppose anybody saw him come on
to the ship—Oh, but Heavens! we haven't got enough
salt. Besides, he'd be sure to taste of tobacco."

I was expecting major changes to this passage, or its complete
removal. I would never have guessed the changes that were actually
made. Here is the revised version of the passage, with the changed
part marked in boldface:

"I don't know what the mischief we're going to do now," I
heard her whisper to Bumpo. "We've no money to buy any more;
and that salt beef was the most important part of the stores."

"Would it not be good political economy," Bumpo whispered
back, "if we salted the able seaman and ate him instead? I
should judge that he would weigh more than a hundred and
twenty pounds."

"Don't be silly,"
snapped Polynesia. "Those things are not done anymore.—Still,"
she murmured after a moment's thought, "it's an
awfully bright idea. I don't suppose anybody saw him come on
to the ship—Oh, but Heavens! we haven't got enough
salt. Besides, he'd be sure to taste of tobacco."

The reference to 'white men' has been removed, but rest of passage,
which I would consider to be among the most potentially offensive of
the entire book, with its association of Bumpo with cannibalism, is
otherwise unchanged. I was amazed. It is interesting to notice that
the references to cannibalism have been excised from a passage on
page 30:

"There were great doings in Jolliginki when he left. He was
scared to death to come. He was the first man from that
country to go abroad. He thought he was going to be eaten by
white cannibals or something.

The revised edition cuts the sentence about white cannibals. The rest
of the paragraph continues:

"You know what those niggers are—that ignorant! Well!—But
his father made him come. He said that all the black kings
were sending their sons to Oxford now. It was the fashion, and
he would have to go. Bumpo wanted to bring his six wives with
him. But the king wouldn't let him do that either. Poor Bumpo
went off in tears—and everybody in the palace was crying
too. You never heard such a hullabaloo."

The revised version reads:

"But his father made him come. He said that all the African
kings were sending their sons to Oxford now. It was the
fashion, and he would have to go. Poor Bumpo went off in
tears—and everybody in the palace was crying too. You never
heard such a hullabaloo."

The six paragraphs that follow this, which refer to the Sleeping
Beauty subplot from the previous book, The Story of Doctor
Dolittle, have been excised. (More about this later.)

There are some apparently trivial changes:

"Listen," said Polynesia, "I've been breaking my head trying to
think up some way we can get money to buy those stores with; and
at last I've got it."

"The money?" said Bumpo.

"No, stupid. The idea—to make the money with."

The revised edition omits 'stupid'. (p.155)
On page 230:

"Poor perishing heathens!" muttered Bumpo. "No wonder the old
chief died of cold!"

becomes

"No wonder the old chief died of cold!" muttered Bumpo.

I gather from other people's remarks that the changes to The
Story of Doctor Dolittle were much more extensive. In
Story (in which Bumpo first appears) there is a subplot
that concerns Bumpo wanting to be made into a white prince. The
doctor agrees to do this in return for help escaping from jail.

When I found out this had been excised, I thought it was unfortunate.
It seems to me that it was easy to view the original plot as a
commentary on the cultural appropriation and racism that accompanies
colonialism. (Bumpo wants to be a white prince because he has become
obsessed with European fairy tales, Sleeping Beauty in
particular.) Perhaps had the book been left intact it might have
sparked discussion of these issues. I'm told that this subplot was
replaced with one in which Bumpo wants the Doctor to turn him into a
lion.

Richter confirms Reed's analysis: By the 18th century, nearly everyone
was reckoning years to start on 1 January except certain official
legal documents. The official change of New Year's day was only to
bring the legal documents into conformance with what everyone was
already doing. So when Franklin's birthdate is reported as 6 January
1706, it means 1706 according to modern reckoning (that is, January
300 years ago) and not 1706 in the "official" reckoning (which would
have been only 299 years ago).

However, Corprew Reed writes to suggest that I am mistaken. Reed
points out that although the legal start of the year prior to
1752 was 25 March, the common usage was to cite 1 January as the start
of the year. The the
British Calendar Act of 1751 even says as much:

WHEREAS the legal Supputation of the Year . . .
according to which the Year beginneth on
the 25th Day of March, hath been found by Experience to be attended
with divers Inconveniencies, . . . as it differs . . . from the common
Usage throughout the whole Kingdom. . .

So Reed suggests that when Franklin (and others) report his birthdate
as being 6 January 1706, they are referring to "common usage", the
winter of the official, legal year 1705, and thus that Franklin really
was born exactly 300 years ago as of Tuesday.

If so, this would be a great relief to me. It was really bothering me
that everyone might be clebrating Franklin's 300th birthday a year
early without realizing it.

I'm going to try to see who here at Penn I can bother about it to find
out for sure one way or the other. Thanks for the suggestion,
Corprew!

Benjamin Franklin was not impressed with the Quakers. His
Autobiography, which is not by any means a long book,
contains at least five stories of Quaker hypocrisy. I remembered only
two, and found the others when I was looking for these.

In one story, the firefighting company was considering contributing
money to the drive to buy guns for the defense of Philadelphia against
the English. A majority of board members was required, but twenty-two
of the thirty board members were Quakers, who would presumably oppose
such an outlay. But when the meeting time came, twenty-one of the
Quakers were mysteriously absent from the meeting! Franklin and his
friends agreed to wait a while to see if any more would arrive, but
instead, a waiter came to report to him that eight of the Quakers were
awaiting in a nearby tavern, willing to come vote in favor of the guns
if necessary, but that they would prefer to remain absent if it
wouldn't affect the vote, "as their voting for such a measure might
embroil them with their elders and friends."

Franklin follows this story with a long discourse on the subterfuges
used by Quakers to pretend that they were not violating their pacifist
principles:

My being many years in the Assembly. . .
gave me frequent opportunities of
seeing the embarrassment given them by their principle against war,
whenever application was made to them, by order of the crown, to grant
aids for military purposes. . . . The common mode at last was, to
grant money under the phrase of its being "for the king's use," and
never to inquire how it was applied.

And a similar story, about a request to the Pennsylvania Assembly for
money to buy gunpowder:

. . . they could not grant money to buy powder, because that was an
ingredient of war; but they voted an aid to New England of three
thousand pounds, to he put into the hands of the governor, and
appropriated it for the purchasing of bread, flour, wheat, or other
grain. Some of the council, desirous of giving the House still further
embarrassment, advis'd the governor not to accept provision, as not
being the thing he had demanded; but he reply'd, "I shall take the
money, for I understand very well their meaning; other grain is
gunpowder," which he accordingly bought, and they never objected to it.

And Franklin repeats an anecdote about William Penn himself:

The honorable and learned Mr. Logan, who had always
been of that sect . . . told me the following anecdote of his old
master, William Penn, respecting defense. It was war-time, and their
ship was chas'd by an armed vessel, suppos'd to be an enemy. Their
captain prepar'd for defense; but told William Penn and his company of
Quakers, that he did not expect their assistance, and they might
retire into the cabin, which they did, except James Logan, who chose
to stay upon deck, and was quarter'd to a gun. The suppos'd enemy
prov'd a friend, so there was no fighting; but when [Logan] went down
to communicate the intelligence, William Penn rebuk'd him severely for
staying upon deck, and undertaking to assist in defending the vessel,
contrary to the principles of Friends, especially as it had not been
required by the captain. This reproof, being before all the company,
piqu'd [Mr. Logan], who answer'd, "I being thy servant, why did thee
not order me to come down? But thee was willing enough that I should
stay and help to fight the ship when thee thought there was
danger."

Clinton Pierce
has provided information which, if true, is probably the answer:

The reason for deleting the 3rd - 13th of September is that in that
span there are no significant Holy Days on the Anglican calendar (at
least that I can tell). September 8th's "Birth of the Blessed Virgin
Mary" is actually an alternate to August 14th. It's also one of the
few places on the 1752 calendar where this empty span occurs beginning
at midweek.

This would also allow the autumnal equinox (one of the significant
events mentioned in the Act) to fall properly on the 21st of September
wheras doing the adjustment in October (the other late 1752 span of no
Holy Days) wouldn't permit that.

If I have time, I will try to dig up an authoritative ecclesiastical
calendar for 1752. The ones I have found online show several other
similar gaps; for example, it seems that 12 January could have been
followed by 24 January, or 14 June followed by 26 June. But these
calendars may not be historically accurate---that is, they may simply
be anachronistically projecting the current practices back to 1752.

Daniel
Dennett is a philosopher of mind and consciousness. The first
work of his that came to my attention was his essay "Why You Can't
Build a Computer That Can Feel Pain". This is just the sort of topic
that college sophomores love to argue about at midnight in the dorm
lounge, the kind of argument that drives me away, screaming "Shut up!
Shut up! Shut up!"

But to my lasting surprise, this essay really had something to say.
Dennett marshaled an impressive amount of factual evidence for his
point of view, and found arguments that I wouldn't have thought of.
At the end, I felt as though I really knew something about this topic,
whereas before I read the essay, I wouldn't have imagined that there
was anything to know about it. Since then, I've tried hard to
read everything I can find that Dennett has written.

I highly recommend Dennett's 1995 book Darwin's Dangerous
Idea. It's a long book, and it's not the main point of my
essay today. I want to give you some sense of what it's about,
without straining myself to write a complete review. Like all really
good books, it has several intertwined themes, and my quoting can only
expose part of one of them:

A teleological explanation is one the explains the existence or
occurrence of something by citing a goal or purpose that is served by
the thing. Artifacts are the most obvious cases; the goal or purpose
of an artifact is the function it was designed to serve by its
creator. There is no controversy about the telos of a hammer:
it is for hammering in and pulling out nails. The telos of
more complicated artifacts, such as camcorders or tow trucks or CT
scanners, is if anything more obvious. But even in simple cases, a
problem can be seen to loom in the background:

"Why are you sawing that board?"
"To make a door."
"And what is the door for?"
"To secure my house."
"And why do you want a secure house?"
"So I can sleep nights."
"And why do you want to sleep nights?"
"Go run along and stop asking such silly questions."

This exchange reveals one of the troubles with teleology: where does
it all stop? What final cause can be cited to bring this
hierarchy of reasons to a close? Aristotle had an answer: God, the
Prime Mover, the for-which to end all for-whiches. The
idea, which is taken up by the Christian, Jewish, and Islamic
traditions, is that all our purposes are ultimately God's
purposes. . . . But what are God's purposes? That is something of a
mystery.

. . . One of Darwin's fundamental contributions is showing us a new
way to make sense of "why" questions. Like it or not, Darwin's idea
offers one way—a clear, cogent, surprisingly versatile way—of
dissolving these old conundrums. It takes some getting used to, and
is often misapplied, even by its staunchest friends. Gradually
exposing and clarifying this way of thinking is a central project of
the present book. Darwinian thinking must be carefully distinguished
from some oversimplified and all-too-popular impostors, and this will
take us into some technicalities, but it is worth it. The prize is,
for the first time, a stable system of explanation that does not go
round and round in circles or spiral off in an infinite regress of
mysteries. Some people would very much prefer the infinite regress of
mysteries, apparently, but in this day and age the cost is
prohibitive: you have to get yourself deceived. You can either
deceive yourself or let others do the dirty work, but there is no
intellectually defensible way of rebuilding the mighty barriers to
comprehension that Darwin smashed.

(Darwin's Dangerous Idea, pp. 24–25.)

Anyway, there's one place in this otherwise excellent book where
Dennett really blew it. First he quotes from a 1988 Boston
Globe article by Chet Raymo, "Mysterious Sleep":

University of Chicago sleep researcher Allan Rechtshaffen asks "how
could natural selection with its irrevocable logic have 'permitted'
the animal kingdom to pay the price of sleep for no good reason?
Sleep is so apparently maladaptive that it is hard to understand why
some other condition did not evolve to satisfy whatever need it is
that sleep satisfies.

And then Dennett argues:

But why does sleep need a "clear biological function" at all? It is
being awake that needs an explanation, and presumably its
explanation is obvious. Animals—unlike plants—need to be awake at
least part of the time in order to search for food and procreate, as
Raymo notes. But once you've headed down this path of leading an
active existence, the cost-benefit analysis of the options that arise
is far from obvious. Being awake is relatively costly, compared with
lying dormant. So presumably Mother Nature economizes where she
can. . . . But surely we animals are at greater risk from predators
while we sleep? Not necessarily. Leaving the den is risky, too, and
if we're going to minimize that risky phase, we might as well keep the
metabolism idling while we bide our time, conserving energy for the
main business of replicating.

This is a terrible argument, because Dennett has apparently missed the
really interesting question here. The question isn't why we sleep;
it's why we need to sleep. Let's consider another important
function, eating. There's no question about why we eat. We eat
because we need to eat, and there's no question about why we
need to eat either. Sure, eating might be maladaptive: you have to
leave the den and expose yourself to danger. It would be very
convenient not to have to eat. But just as clearly, not eating won't
work, because you need to eat. You have to get energy from somewhere;
you simply cannot run your physiology without eating something once in
a while. Fine.

But suppose you are in your den, and you are hungry, and need to go
out to find food. But there is a predator sniffing around the door,
waiting for you. You have a choice: you can stay in and go hungry,
using up the reserves that were stored either in your body or in your
den. When you run out of food, you can still go without, even though
the consequences to your health of this choice may be terrible. In
the final extremity, you have the option of starving to death, and
that might, under certain circumstances, be a better strategy than
going out to be immediately mauled by the predator.

With sleep, you have no such options. If you're treed by a panther,
and you need to stay awake to balance on your branch, you have no
options. You cannot use up your stored reserves of sleep. You do not
have the option to go without sleep in the hope that the panther will
get bored and depart. You cannot postpone sleep and suffer the
physical consequences. You cannot choose to die from lack of sleep
rather than give up and fall out of the tree. Sooner or later you
will sleep, whether you choose to or not, and when you sleep you will
fall out of the tree and die.

People can and do go on hunger strikes, refuse to eat, and starve to
death. Nobody goes on sleep strikes. They can't. Why not? Because
they can't. But why can't they? I don't think anyone
knows.

The question isn't about the maladaptivity of sleeping itself; it's
about the maladaptivity of being unable to prevent or even to
delay sleep. Sleep is not merely a strategy to keep us
conveniently out of trouble. If that were all it was, we would need
to sleep only when it was safe, and we would be able to forgo it when
we were in trouble. Sleep, even more than food, must serve some vital
physiological role. The role must be so essential that it is
impossible to run a mammalian physiology without it, even for as long
as three days. Otherwise, there would be adaptive value in being able
to postpone sleep for three days, rather than to fall asleep
involuntarily and be at the mercy of one's enemies.

Given that, it is indeed a puzzle that we have not been able to
identify the vital physiological role of sleep, and Rechtshaffen's
puzzlement above makes sense.

An adjustment to Franklin's birthday
Thanks to the wonders of the Internet, the
text of the British Calendar Act of 1751 is available. (Should
you read the Act, it may be helpful to know that the obscure word
"supputation" just means "calculation".) This is the act that
adjusted the calendar from Julian to Gregorian and fixed the 11-day
discrepancy that had accumulated since the Nicean Council in 325 CE,
by deleting September 3-13, so that the month of September 1752 had
only 19 days:

September 1752

Sun

Mon

Tue

Wed

Thu

Fri

Sat

1

2

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Why September 3-13? I don't know, although I would love to find out.
There are at least two questions here: Why start on the third of the month?
Clearly you don't want to delete either the first or the last day of
the month, because all sorts of things are scheduled to occur on those
days, and deleting them would cause even more confusion than would
deleting the middle days. But why not delete the second through the
twelfth?

And why September? Had I been writing the Act, I think I would have
preferred to delete a chunk of February; nobody likes February
anyway.

Anyway, the effect of this was to make the year 1752 only 355 days
long, instead of the usual 366.

I hadn't remembered, however, that this act was also the one that
moved the beginning of the year from 25 March to 1 January. Since
1752 was the first civil year to begin on 1 January, that meant that
1751 was only 282 days long, running from 25 March through 31
December. I used to think that the authors of the Unix cal
program were very clever for getting September 1752 correct:

This is quite wrong, since 1751 started on March 25, and there was no
such thing as January 1751 or February 1751.

When you excise eleven days from the calendar, you have a lot of
puzzles. For any event that was previously scheduled to occur on or
after 14 September, 1752, you now need to ask the question: should you
leave its nominal date unchanged, so that the event actually occurs 11
days sooner than it would have, or do you advance its nominal date 11
days forward? The Calendar Act deals with this in some detail.
Certain court dates and ecclesiastical feasts, including corporate
elections, are moved forward by 11 real days, so that their nominal
dates remain the same; other events are adjusted so that the occur at
the same real times as they would have without the tamperings of the
calendar act. Private functions are not addressed; I suppose the
details were left up to the convenience of the participants.

Historians of that period have to suffer all sorts of annoyances in
dealing with the dates, since, for example, you find English accounts
of the Battle of Gravelines occurring on 28 July, but Spanish accounts
that their Armada wasn't even in sight of Cornwall until 29 July.
Sometimes the histories will use a notation like "11/21 July" to mean
that it was the day known as 11 July in England and 21 July in Spain.
I find this clear, but the historians mostly seem to hate this
notation. ("Fractions! If I wanted to deal in fractions, I would have
become a grocer, not a historian!")

You sometimes hear that there were riots by tenants, angry to be
paying a full month's rent for only 19 days of tenancy in September
1752. I think this is a myth. The act says quite clearly:

. . . nothing in this present Act contained shall extend, or be construed to
extend, to accelerate or anticipate the Time of Payment of any Rent or
Rents, Annuity or Annuities, or Sum or Sums of Money whatsoever. . .
or the Time of doing any Matter or Thing directed or required by any
such Act or Acts of Parliament to be done in relation thereto; or to
accelerate the Payment of, or increase the Interest of, any such Sum
of Money which shall become payable as aforesaid; or to accelerate the
Time of the Delivery of any Goods, Chattles, Wares, Merchandize or
other Things whatsoever . . .

It goes on in that vein for quite a while, and in particular, it says
that "all and every such Rent and Rents. . . shall remain and continue
to be due and payable; at and upon the same respective natural Days
and Times, as the same should and ought to have been payable or made,
or would have happened, in case this Act had not been made. . . ". It
also specifies that interest payments are to be reckoned according to
the natural number of days elapsed, not according to the calendar
dates. There is also a special clause asserting that no person shall
be deemed to have reached the age of twenty-one years until they are
actually twenty-one years old, calendrical trickery
notwithstanding.

I first
brought this up in connection with Benjamin Franklin's 300th
birthday, saying that although Franklin had been born on 6
January, 1706, his birthday had been moved up 11 days by the Act. But
things seem less clear to me now that I have reread the act. I
thought there was a clause that specifically moved birthdays forward,
but there isn't. There is the clause that says that Franklin
cannot be said to be 300 years old until 17 January, and it also says
that dates of delivery of merchandise should remain on the same
real days. If you had contracted for flowers
and cake to be delivered to a birthday party to be held on 6 January
2006, the date of delivery is advanced so that the florist and the
baker have the same real amount of time to make delivery, and are now
required to deliver on 17 January 2006.

But there is the additional confusion I had forgotten, which is that
Franklin was born on 6 January 1706, and there was no 6 January
1751. What would have been 6 January 1751 was renominated to be
6 January 1752 instead, and then the old 6 January 1752 was
renominated as 17 January 1753.

To make the problem more explicit, consider John Smith, born 1 January
1750. The previous day was 31 December 1750, not 1749, because 1749
ended nine months earlier, on March 24. Similarly, 1751 will not
begin until 25 March, when John is 84 days old. 1751 is an oddity,
and ends on December 31, when John is 364 days old. The following day
is 1 January 1752, and John is now one year old. Did you catch that?
John was born on 1 January 1750, but he is one year old on 1 January
1752. Similarly, he is two years old
on 1 January 1753.

The same thing
happens with Benjamin Franklin.
Franklin was born on 6 January 1706, so he will be 300 years old (that
is, 365 × 300 + 73 = 109573 days old) on 17 January 2007.

So I conclude that the cake and flowers for Franklin's 300th birthday
celebration are being delivered a year early!

Franklin was born on 6 January, 1706. When they switched to the
Gregorian calendar in 1752, everyone had their birthday moved forward
eleven days, so Franklin's moved up to 17 January. (You need to do
this so that, for example, someone who is entitled to receive a trust
fund when he is thirty years old does not get access to it eleven days
before he should. This adjustment is also why George Washington's
birthday is on 22 February even though he was born 11 February 1732.)

(You sometimes hear claims that there were riots when the calendar was
changed, from tenants who were angry at paying a month's rent for only
19 days of tenancy. It's not true. The English weren't stupid. The
law that adjusted the calendar specified that monthly rents and such
like would be pro-rated for the actual number of days.)

Since I live in Philadelphia, Franklin is often in my thoughts. In
the 18th century, Franklin was Philadelphia's most important citizen.
(When I first moved here, my girlfriend of the time sourly observed
that he was still Philadelphia's most important citizen.
Philadelphia's importance has faded since the 18th century, leaving it
with a forlorn nostalgia for Colonial days.) When you read
Franklin's Autobiography, you hear him discussing places
in the city that are still there:

So not considering or knowing the difference of money, and the greater
cheapness nor the names of his bread, I made him give me three-penny
worth of any sort. He gave me, accordingly, three great puffy rolls. I
was surpriz'd at the quantity, but took it, and, having no room in my
pockets, walk'd off with a roll under each arm, and eating the other.

Thus I went up Market-street as far as Fourth-street, passing by the
door of Mr. Read, my future wife's father; when she, standing at the
door, saw me, and thought I made, as I certainly did, a most awkward,
ridiculous appearance.

Heck, I was down at Fourth and Market just last month.

Franklin's personality comes across so clearly in his
Autobiography and other writings that it's easy to
imagine what he might have been like to talk to.
I sometimes like to pretend that Franklin and I are walking around
Philadelphia together. Wouldn't he be surprised at what Philadelphia
looks like, 250 years on! What questions does Franklin have? I spend
a lot of time explaining to Franklin how the technology works.
(People who pass me in the street probably think I'm insane, or else
that I'm on the phone.) Some of the explaining is easy, some less so.
Explaining how cars work is easy. Explaining how cell phones work is
much harder.

Here's my favorite quotation from Franklin:

I believe I have omitted mentioning that, in my first
voyage from Boston, being becalm'd off Block Island, our people set
about catching cod, and hauled up a great many. Hitherto I had stuck
to my resolution of not eating animal food, and on this occasion
consider'd, with my master Tryon, the taking every fish as a kind of
unprovoked murder, since none of them had, or ever could do us any
injury that might justify the slaughter. All this seemed very
reasonable. But I had formerly been a great lover of fish, and, when
this came hot out of the frying-pan, it smelt admirably well. I
balanc'd some time between principle and inclination, till I
recollected that, when the fish were opened, I saw smaller fish taken
out of their stomachs; then thought I, "If you eat one another, I
don't see why we mayn't eat you." So I din'd upon cod very heartily,
and continued to eat with other people, returning only now and then
occasionally to a vegetable diet. So convenient a thing it is to be a
reasonable creature, since it enables one to find or make a reason for
everything one has a mind to do.

It's with some trepidation that I'm starting this section of my
blog, because I really don't understand physics very well. Most of
it, I suspect, will be about how little I understand. But I'm going
to give it a whirl.

Sometime this summer it occurred to me that the phenomenon of
transparency is more more complex than one might initially think, and
more so than most people realize. I went around asking people with
more physics education than I had if they could explain to me why
things like glass and water are transparent, and it seemed to me that
not only did they not understand it any better than I did, but they
didn't realize that they didn't understand it.

A common response, for example, was that media like glass and water
are transparent because the light passes through them unimpeded. This
is clearly wrong. We can see that it's wrong because both glass and
water have a tendency to refract incident light. Unimpeded photons
always travel in straight lines. If light is refracted by a medium,
it is because the photons couple with electrons in the medium, are
absorbed, and then new photons are emitted later, going in a different
direction. So the photons are being absorbed; they are not passing
through the water unimpeded. (Similarly, light passes more slowly
though glass and water than it does through vacuum, because of the
time taken up by the interactions between the photons and the
electrons. If the photons were unimpeded by electromagnetic effects,
they would pass through with speed c.)

Sometimes the physics students tell me that some of the photons
interact, but the rest pass through unimpeded. This is not the case
either. If some of the photons were unimpeded when passing through
water or glass, then you would see two images of the other side: one
refracted, and one not. But you don't; you see only one image. (Some
photons are reflected completely, so you do see two images: a
reflected one, and a transmitted one. But if photons were refracted
internally as well as being transmitted unimpeded, there would be two
transmitted images.) This demonstrates that all the photons
are interacting with the medium.

The no-interference explanation is correct for a vacuum, of
course. Vacuum is transparent because the photons pass through it
with no interaction. So there are actually two separate phenomena
that both go by the name of "transparency". The way in which vacuum
is transparent is physically different from the way in which glass is
transparent. I don't know which of these phenomena is responsible for
the transparency of air.

Now, here is the thing that was really puzzling me about glass and
water. For transparency, you need the photons to come out of the
medium going in the same direction as they went in. If photons are
scattered in all different directions, you get a translucent or opaque
medium. Transparency is only achieved when the photons that come out
have the same velocity and frequency as those going in, or at least
when the outgoing velocity depends in some simple fashion on the
incoming velocity, as with a lens.

Since the photon that comes out of glass is going in the exact same
direction as the photon that went in, something very interesting is
happening inside the glass. The photon reaches the glass and is
immediately absorbed by an electron. Sometime later, the electron
emits a new photon. The new photon is travelling in exactly the same
direction as the old photon. The new photon is absorbed by the next
electron, which later emits another photon, again travelling in the
exact same direction. This process repeats billions of times until
the final photon is ejected on the other side of the glass, still in
the exact same direction, and goes on its way.

Even a tiny amount of random scattering of the photons would
disrupt the transparency completely. So, I thought, I would expect to
find transparency in media that have a very rigid crystalline
structure, so that the electromagnetic interactions would be exactly
the same at each step. But in fact we find transparency not in
crystalline substances, such as metals, but rather in amorphous ones,
like glass and water! This was the big puzzle for me. How can it be
that the photon always comes out in the same direction that it went
in, even though it was wandering about inside of glass or water, which
have random internal structures?

Nobody has been able to give me a convincing explanation of this.
After much pondering, I have decided that it is probably due to
conservation of relativistic momentum. Because of quantum
constraints, the electron can only emit a new photon of the same
energy as the incoming photon. Because the electron is bound in an
atom, its own momentum cannot change, so conservation of momentum
requires that the outgoing photon have the same velocity as the
incoming one.

I would like to have this confirmed by someone who really
understands physics. So far, though, I haven't met anyone like that!
I'm thinking that I should start attending physics colloquia at the
University of Pennsylvania and see if I can buttonhole a couple of
solid-state physics professors. Even if I don't buttonhole anyone,
going to the colloquia might be useful and interesting. I should
write a blog post about why it's easier to learn stuff from a
colloquium than from a book.

Medieval Chinese typesetting technique
One of my longtime fantasies has been to write a book called
Quipus and Abacuses: Digital Information Processing Before
1946. The point being that digital information processing did
exist well before 1946, when large-scale general-purpose electronic
digital computers first appeared. (Abacuses you already know about,
and a future blog posting may discuss the use of abacuses in Roman
times and in medieval Europe. Quipus are bunches of knotted cords
used in Peru to record numbers.)
There are all sorts of interesting questions to be answered. For
instance, who first invented alphabetization? (Answer: the scribes at
the Great Library in Alexandria, around 200 CE.) And how did they do
it? (Answer to come in a future blog posting.) How were secret
messages sent? (Answer: lots of steganography.) How did people do
simple arithmetic with crappy Roman numerals? (Answer: abacuses.)
How were large quantities of records kept, indexed, and searched? How
were receipts made when the recipients were illiterate?

Here's a nice example. You may have heard that the Koreans and the
Chinese had printing presses with movable type before Gutenberg
invented it in Europe. How did they organize the types?

In Europe, there is no problem to solve. You have 26 different types
for capital letters and 26 for small letters, so you make two type
cases, each divided into 26 compartments. You put the capital letter
types in the upper case and the small letter types in the lower case.
(Hence the names "uppercase letter" and "lowercase letter".) You put
some extra compartments into the cases for digits, punctuation
symbols, and blank spaces. When you break down a page, you sort the
types into the appropriate compartments. There are only about 100
different types, so whatever you do will be pretty easy.

However, if you are typesetting Chinese, you have a much bigger
problem on your hands. You need to prepare several thousand types
just for the common characters. You need to store them somehow, and
when you are making up a page to be printed you need to find the
required types efficiently. The page may require some rare
characters, and you either need to have up to 30,000 rarely-used types
made up in advance or some way to quickly make new types as needed.
And you need a way to sort out the types and put them away in order
when the page is complete.

(I'm sure some reader is itching to point out that Korean is written with a
phonetic alphabet, hangul, which avoids the problem by having
only 28 letters. But in fact that is wrong for two reasons. First,
the layout of Korean writing requires that a type be made for each two-
or three-letter syllable. And second, perhaps more to the point,
moveable type presses were used in Korea before the invention of
hangul, before Korean even had a written form. Movable type
was invented in Korea around 1234 CE; hangul was first
promulgated by Sejong the Great in 1443 or 1444. The first Korean
moveable type presses were used to typeset documents in Chinese, which
was the language of scholarship and culture in Korea until the 19th
century.)

In fact, several different solutions were adopted. The earliest
movable types in China were made of clay mixed with glue. These had
the benefit of being cheap. Copper types were made later, but had two
serious disadvantages. First, they were very expensive. And second,
since much of their value could be recovered by melting them down, the
government was always tempted to destroy them to recover the copper,
which did indeed happen.

Wang
Chen (王禎), in 1313, writes that the types were organized as follows:
There were two circular bamboo tables, each seven feet across and with
one leg in the middle; the tabletops were mounted on the legs so that
they could rotate. One table was for common types and the other for
the rare, one-off types. The top of each table was divided into eight
sections, and in each section, types were arranged in their numerical
order according to their listing in the Book of Rhymes, an early
Chinese dictionary that organized the characters by their sounds.

To set the type for a page, the compositors would go through the proof
and number each character with a code indicating its code number from
the Book of Rhymes. One compositor would then read from the list of
numbers while the other, perched on a seat between the two rotating
tables, would select the types from the tables. Wang doesn't say, but
one supposes that the compositors would first put the code numbers
into increasing order before starting the search for the right types.
This would have two benefits: First, it would enable a single pass to
be made over the two tables, and second, if a certain character
appeared multiple times on the page, it would allow all the types
needed for that character to be picked up at once.

The types would then be inserted into the composition frame. If a
character was needed for which there was no type, one was made on the
spot. Wang Chen's types were made of wood. The character was
carefully written on very thin paper, which was then pasted
upside-down onto a blank type slug. A wood carver with a delicate
chisel would then cut around the character into the wood.

In 1776 a great printing project was overseen by Jian Jin (Chin
Ch'ien), also using wooden types. Jin left detailed instructions
about how the whole thing was accomplished. By this time the Book of
Rhymes had been superseded.

The Imperial
K'ang Hsi Dictionary (K'ang-hsi tzu-tien or Kāngxī
Zìdiǎn, 康熙字典), written between 1710 and 1716,
was the gold standard for Chinese dictionaries at the time, and to
some extent, still is, since it set the pattern for the organization
of Chinese characters that is still followed today. If you go into a
store and buy a Chinese dictionary (or a Chinese-English dictionary)
that was published last week, its organization will be essentially the
same as that of the Imperial K'ang Hsi Dictionary. Since readers may
be unfamiliar with the organization of Chinese dictionaries, I will
try to explain.

Characters are organized primarily by a "classifier", more usually
called a "radical" today. The typical Chinese character incorporates
some subcharacters. For example, the character for "bright" is
clearly made up of the characters for "sun" and "moon"; the character
for "sweat" is made up of "water" and "shield". (The "shield" part is
not because of anything relating to a shield, but because it
sounds like the word for "shield".) Part of each character is
designated its radical. For "sweat", the radical is "water"; for
"bright" it is "sun". How do you know that the radical for "bright"
is "sun" and not "moon"? You just have to know.

What about characters that are not so clearly divisible? They have
radicals too, some of which were arbitrarily designated long ago, and
some of which were designated based on incorrect theories of
etymology. So some of it is arbitrary. But all ordering of words is
arbitrary to some extent or another. Why does "D" come before "N"?
No reason; you just have to know. And if you have ever seen a
first-grader trying to look up "scissors" in the dictionary, you know
how difficult it can be. How do you know it has a "c"? You just have
to know.

Anyway, a character has a radical, which you can usually guess in at
most two or three tries if you don't already know it. There are
probably a couple of hundred radicals in all, and they are ordered
first by number of strokes, and then in an arbitrary but standard
order among the ones with the same number of strokes. The characters
in the dictionary are listed in order by their radical. Then, among
all the characters with a particular radical, the characters are
ordered by the number of strokes used in writing them, from least to
most. This you can tell just by looking at the characters. Finally,
among characters with the same number of strokes and the same radical,
the order is arbitrary. But it is still standardized, because it is
the order used by the Imperial K'ang Hsi Dictionary.

So if you want to look up some character like "sweat", you first
identify the radical, which is "water", and has four strokes. You
look in the radical index among the four-stroke radicals, of which
there are no more than a couple dozen, until you find "water", and
this refers you to the section of the dictionary where all the
characters with the "water" radical are listed. You turn to this
section, and look through it for the subsection for characters that
have seven strokes. Among these characters, you search until you find
the one you want.

This is the solution to the problem of devising an ordering for the
characters in the dictionary. Since this ordering was (and is)
well-known, Jin used it to organize his type cases. He writes:

Label and arrange twelve wooden cabinets according to the
names of the twelve divisions of the Imperial K'ang Hsi
Dictionary. The cabinets are 5'7" high, 5'1" wide, 2'2" deep
with legs 1'5" high. Before each one place a wooden bench of
the same height as the cabinet's legs; they are convenient to
stand on when selecting type. Each case has 200 sliding
drawers, and each drawer is divided into eight large and eight
small compartments, each containing four large or four small
type. Write the characters, with their classifiers and number
of strokes, on labels on the front of each drawer.

When selecting type, first examine the make-up of the
character for its corresponding classifier, and then you will
know in which case it is stored. Next, count the number of
strokes, and then you will know in which drawer it is. If one
is experienced in this method, the hand will not err in its
movements.

There are some rare characters that are seldom used, and for
which few type will have been prepared. Arrange small
separate cabinets for them, according to the twelve divisions
mentioned above, and place them on top of each type case where
they may be seen at a glance.

The size measurements here are misleading. The translator says that
the "inch" used here is the Chinese inch of the time, which is about
32.5 mm, not the 25.4 mm of the modern inch. He does not say what is
meant by "foot"; I assume 12 inches. That means that the type cases
are actually 7'2" high, 6'6" wide, 2'9" deep, (218 cm × 198 cm
× 84 cm) with legs 1'10" high (55 cm), in modern units.

(Addendum 20060116: The quote doesn't say, but the illustration in Jin's book
shows that the cabinets have 20 rows of 10 drawers each.)

One puzzle I have not resolved is that there do not appear to be
enough type drawers. Jin writes that there are twelve cabinets with
200 drawers each; each drawer contains 16 compartments, and each
compartment four type. This is enough space for 153,600 types
(remember that you need multiples of the common characters), but Jin
reports that 250,000 types were cut for his project. Still, it seems
clear that the technique is feasible.

Another puzzle is that I still don't know what the "twelve divisions"
of the Imperial K'ang Hsi Dictionary are. I examined a copy in the
library and I didn't see any twelve divisions. Perhaps some reader
can enlighten me.

As in Wang's project, one compositor would first go over the proof
page, making a list of which types needed to be selected, and how
many; new types were cut from wood as needed. Then compositors would
visit the appropriate cases to select the types as necessary; another
compositor would set the type, and then the page would be printed, and
the type broken down. These activities were always going on in
parallel, so that page n was being printed while page
n+1 was being typeset, the types for page n+2 were being
selected, and page n-1 was being broken down and its types
returned to the cabinet.

Since I mentioned the book Liber Abaci, written in 1202
by Leonardo Pisano (better known as Fibonacci) in an earlier post, I
may as well quote you its most famous passage:

A certain man had one pair of rabbits together in a certain
enclosed place, and one wishes to know how many are created
from the pair in one year when it is the nature of them in a
single month to bear another pair, and in the second month
those born to bear also.

Because the abovewritten pair in the first month bore, you
will double it; there will be two pairs in one month. One of
these, namely the first, bears in the second month, and thus
there are in the second month 3 pairs; of these in on month
two are pregnant, and in the third month 2 pairs of rabbits
are born, and thus there are 5 pairs in the month...

You can indeed see in the margin how we operated, namely that
we added the first number to the second, namely the 1 to the
2, and the second to the third, and the third to the fourth,
and the fourth to the fifth, and thus one after another until
we added the tenth to the eleventh, namely the 144 to the 233,
and we had the abovewritten sum of rabbits, namely 377, and
thus you can in order find it for an unending number of
months.

This passage is the reason that the Fibonacci numbers are so called.

Much of Liber Abaci is incredibly dull, with pages and
pages of stuff of the form "Then because three from five is two, you
put a two under the three, and then because eight is less than six you
take the four that is next to the six and make forty-six, and take
eight from forty-six and that is thirty-eight, so you put the eight
from the thirty-eight under the six...". And oh, gosh, it's just
deadly. I had hoped I might learn something interesting about the way
people did arithmetic in 1202, but it turns out it's almost exactly
the same as the way we do it today.

But there's some fun stuff too. In the section just before the one
about the famous rabbits, he presents (without proof) the formula
2n-1 · (2n -1) for perfect
numbers. Euler proved in the 18th century that all even perfect
numbers have this form. It's still unknown whether there are any odd
perfect numbers.

Elsewhere, Leonardo considers a problem in which seven men are going
to Rome, and each has seven sacks, each of which contains seven loaves
of bread, each of which is pierced with seven knives, each of which
has seven scabbards, and asks for the total amount of stuff going to
Rome.

Negative numbers weren't widely used in Europe until the 16th
century, but Liber Abaci does consider several problems
whose solution requires the use of negative numbers, and Leonardo
seems to fully appreciate their behavior.

Some sources say that Leonardo was only able to understand
negative numbers as a financial loss; for example Dr. Math says:

Fibonacci, about 1200, allowed negative solutions in financial
problems where they could be interpreted as a loss rather than a
gain.

This, however, is untrue. Understanding a negative number as a loss;
that is, as a relative decrease from one value to another over time,
is a much less subtle idea than to understand a negative number as an
absolute quantity in itself, and it is in the latter way that
Leonardo seems to have understood negative numbers.

In Liber Abaci, Leonardo considers the solution of the following
system of simultaneous equations:

A + P = 2(B + C)B + P = 3(C + D)C + P = 4(D + A)D + P = 5(A + B)

(Note that although there are only four equations for the five
unknowns, the four equations do determine the relative proportions of
the five unknowns, and so the problem makes sense because all the
solutions are equivalent under a change of units.)

Leonardo presents the problem as follows:

Also there are four men; the first with the purse has double
the second and third, the second with the purse has triple the
third and fourth; the third with the purse has quadruple the
fourth and first. The fourth similarly with the purse has
quintuple the first and second;

and then asserts (correctly) that the problem cannot be solved with
positive numbers only:

this problem is not solvable unless it is conceded that the
first man can have a debit,

and then presents the solution:

and thus in smallest numbers the second has 4, the third 1,
the fourth 4, and the purse 11, and the debit of the first man
is 1;

That is, the solution has B=4, C=1, D=4, P=11, and A= -1.

Leonardo also demonstrates understanding of how negative numbers
participate in arithmetic operations:

and thus the first with the purse has 10, namely double the
second and third;

That is, -1 + 11 = 2 · (1 + 4);

also the second with the purse has 15, namely triple the third
and fourth; and the third with the purse has quadruple the
fourth and the first, because if from the 4 that the fourth
man has is subtracted the debit of the first, then there will
remain 3, and this many is said to be had between the fourth
and first men.

The explanation of the problem goes on at considerable length, at least
two full pages in the original, including such observations as:

Therefore the second's denari and the fourth's denari are the
sum of the denari of the four men; this is inconsistent unless
one of the others, namely the first or third has a debit which
will be equal to the capital of the other, because their
capital is added to the second and fourth's denari; and from
this sum is subtracted the debit of the other, undoubtedly
there will remain the sum of the second and fourth's denari,
that is the sum of the denari of the four men."

That is, he reasons that A + B + C + D = B + D, and so therefore A = -C.

Quotations are from Fibonacci's Liber Abaci: A
Translation into Modern English of Leonardo Pisano's Book of
Calculation, L. E. Sigler. Springer, 2002. pp. 484-486.

At OSCON this summer I was talking to Peter Scott (author of Perl Debugged, Perl Medic, and
other books I wanted to write but didn't), and he observed that the
preface of HOP
did not contain a section that explained that the prose text
was on proportional font and the code was all in monospaced font.

I don't remember what (if any) conclusion Peter drew from this, but I
was struck by it, because I had been thinking about that myself for a
couple of days. Really, what is this section for? Does anyone really
need it? Here, for example, is the corresponding section from
Mastering Algorithms with Perl, because it is the first book I
pulled off my shelf:

Conventions Used in This Book

Italic

Used for filenames, directory names, URLs, and occasional
emphasis.

Constant width

Used for elements of programming languages, text manipulated
by programs, code examples, and output.

Constant width bold

Used for use input and for emphasis in code

Constant width italic

Used for replaceable values

Several questions came to my mind as I transcribed that, even though
it was 4 AM.

First, does anyone really read this section and pay attention to it,
making a careful note that italic font is used for filenames,
directory names, URLs, and occasional emphasis? Does anyone, reading
the book, and encountering italic text, say to themselves "I wonder
what the funny font is about? Oh! I remember that note in the
preface that italic font would be used for filenames, directory names,
URLs, and occasional emphasis. I guess this must be one of those."

Second, does anyone really need such an explanation? Wouldn't
absolutely everyone be able to identify filenames, directory names,
URLs, and occasonal emphasis, since these are in italics, without the
explicit directions?

I wonder, if anyone really needed these instructions, wouldn't they be
confused by the reference to "constant-width italic", which isn't
italic at all? (It's slanted, not italic.)

Even if someone needs to be told that constant-width fonts are used
for code, do they really need to be told that constant-width bold
fonts are used for emphasis in code? If so, shouldn't they also be
told that bold roman fonts are used for emphasis in running text?

Some books, like Common Lisp: The Language, have extensive
introductions explaining their complex notational conventions. For
example, pages 4--11 include the following notices:

The symbol "⇒" is used in examples to indicate evaluation.
For example,

(+ 4 5) ⇒ 9

means "the result of evaluating the code (+ 4 5) is (or would
be, or would have been) 9."

The symbol "→" is used in examples to indicate macro
expansions. ...

Explanation of this sort of unusual notation does seem to me to be
valuable. But really the explanations in most computer books make me
think of long-playing record albums that have a recorded voice at the
end of the first side that instructs the listener "Now turn the record
over and listen to the other side."

I don't think omitted this section from HOP on purpose; it simply
never occurred to me to put one in. Had MK asked me about it, I don't
know what I would have said; they didn't ask.

HOP does have at least one unusual typographic
convention: when two versions of the same code are shown, the code in
the second version that was modified or changed has been set in
boldface. I had been wondering for a couple of weeks before OSCON if
I had actually explained that; after running into Peter I finally
remembed to check. The answer: no, there is no explanation. And I
don't think it's a common convention.

But of all the people who have seen it, including a bunch of official
technical reviewers, a few hundred casual readers on the mailing list,
and now a few thousand customers, nobody suggested than an explanation
was needed, and nobody has reported being puzzled. People seem to
understand it right away.

I don't know what to conclude from this yet, although I suspect it will
be something like:

(a) the typographic conventions in typical computer books are
sufficiently well-established, sufficiently obvious, or
both, that you don't have to bother explaining them unles
they're really bizarre,

or:

(b) readers are smarter and more resilient than a lot of
people give them credit for.

Explanation (b) reminds me of a related topic, which is that
conference tutorial attendees are smarter and more resilient than a
lot of conference tutorial speakers give them credit for. I suppose
that is a topic for a future blog entry.

(Consensus on my mailing list, where this was originally posted, was
that the ubiquitous explanations of typographic conventions are not
useful. Of course, people for whom they would have been useful were
unlikely to be subscribers to my mailing list, so I'm not sure we can
conclude anything useful from this.)

Last fall I read a bunch of books on logic and foundations of
mathematics that had been written around the end of the 19th and
beginning of the 20th centuries. I also read some later commentary on
this work, by people like W. V. O. Quine. What follows are some notes
I wrote up afterwards.

The following are only my vague and uninformed impressions. They
should not be construed as statements of fact. They are also poorly
edited.

1. Frege and Peano were the pioneers of modern mathematical logic.
All the work before Peano has a distinctly medieval flavor. Even
transitionary figures like Boole seem to belong more to the old
traditions than to the new. The notation we use today was all
invented by Frege and Peano. Frege and Peano were the first to
recognize that one must distinguish between x and {x}.

(Added later: I finally realized today what this reminds me of. In
physics, there is a fairly sharp demarcation between classical physics
(pre-1900, approximately) and modern physics (post-1900). There was a
series of major advances in physics around this time, in which the old
ideas, old outlooks, and old approaches were swept away and replaced
with the new quantum theories of Planck and Einstein, leaving the
field completely different than it was before. Peano and Frege are
the Planck and Einstein of mathematical logic.)

2. Russell's paradox has become trite, but I think we may have
forgotten how shocked and horrified everyone was when it first
appeared. Some of the stories about it are hair-raising. For example,
Frege had published volume I of his Grundgesetze der Arithmetik
("Basic Laws of Arithmetic"). Russell sent him a letter as volume II
was in press, pointing out that Frege's axioms were inconsistent.
Frege was able to add an appendix to volume II, including a
heartbreaking note:

"Hardly anything more unwelcome can befall a scientific writer than
that one of the foundations of his edifice be shaken after the work is
finished. I have been placed in this position by a letter of
Mr Bertrand Russell just as the printing of the second volume was nearing
completion..."

I hope nothing like this ever happens to any of my dear readers.

The struggle to figure out Russell's paradox took years. It's so
tempting to think that the paradox is just a fluke or a wart. Frege,
for example, first tried to fix his axioms by simply forbidding
(x ∈ x). This, of course, is insufficient, and the
Russell paradox runs extremely deep, infecting not just set theory,
but any system that attempts to deal with properties and
descriptions of things. (Expect a future blog post about this.)

3. Straightening out Russell's paradox went in several different
directions. Russell, famously, invented the so-called "Theory of
Types", presented as an appendix to Principia
Mathematica. The theory of types is noted for being
complicated and obscure, and there were several later simplifications.
Another direction was Zermelo's, which suffers from different defects:
all of Zermelo's classes are small, there aren't very many of them,
and they aren't very interesting. A third direction is von Neumann's:
any sets that would cause paradoxes are blackballed and forbidden from
being elements of other sets.

To someone like me, who grew up on Zermelo-Fraenkel, a term like
"(z = complement({w}))" is weird and slightly uncanny.

(Addendum 20060110: Quine's "New Foundations" program is yet another
technique, sort of a simplified and streamlined version of the theory
of types. Yet another technique, quite different from the others, is
to forbid the use of the ∼ ("not") operator in set comprehensions.
This last is very unusual.)

4. Notation seems to have undergone several revisions since the first
half of the 20th Century. Principia Mathematica and
other works use a "dots" notation instead of or in additional to using
parentheses for grouping. For example, instead of writing "((a
+ b) × c) + ((e + f) × g)",
one would write "a + b .× c :+: e +
f .× g". (This notation was invented by—guess
who?—Peano.) This takes some getting used to when you have not seen
it before. The dot notation seems to have fallen completely out of
use. Last week, I thought it had technical advantages over
parentheses; now I am not sure.

The upside-down-A (∀) symbol meaning "for each" is of more
recent invention than is the upside-down-E (∃) symbol meaning
"there exists". Early C20 would write "∃z:P(z)" as
"(∃z)P(z)" but would write "∀z:
P(z)" as simply "(z)P(z)".

The turnstile symbol $$\vdash$$ is
Russell and Whitehead's abbreviation of the elaborate notation of
Frege's Begriffschrift. The Begriffschrift
notation was essentially annotated abstract syntax trees. The root of
the tree was decorated with a vertical bar to indicate that the
statement was asserted to be true. When you throw away the tree,
leaving only the root with its bar, you get a turnstile symbol.

The ∨ symbol is used for disjunction, but its conjunctive
counterpart, the ∧, is not used. Early C20 logicians use a dot
for conjunction. I have been told that the ∨ was chosen by Russell
and Whitehead as an abbreviation for the Latin vel = "or".
Quine says that the $$\sim$$ denotes logical
negation because of its resemblance to the letter "N" (for "not").
Incidentally, Quine also says that the ↓ that is sometimes used
to mean logical nor is simply the ∨ with a vertical slash through
it, analogous to ≠.

An ι is prepended to an expression x to denote the set
that we would write today as {x}. The set { u :
P(u) } of all u such that P(u) is
true is written as ûP. Peter Norvig says (in
Paradigms of Artificial Intelligence Programming) that
this circumflex is the ultimate source of the use of "lambda" for
function abstraction in Lisp and elsewhere.

5. (Addendum 20060110: Everyone always talks about Russell and
Whitehead's Principia Mathematica, but it isn't; it's
Whitehead and Russell's. Addendum 20070913: In a later article, I
asked how and when Whitehead lost top billing in casual citation;
my conclusion was that it occurred on 10 December, 1950.)

6. (Addendum 20060116: The ¬ symbol is probably an abbreviated version
of Frege's notation for logical negation, which is to attach a little stem
to the underside of the branch of the abstract syntax tree that is to be
negated. The universal quantifier notation current in Principia
Mathematica, to write (x)P(x) to mean that
P(x) is true for all x, may also be an adaptation of
Frege's notation, which is to put a little cup in the branch of the tree
to the left of P(x) and write x in the cup.

[I sent this out to my book discussion mailing
list back in November, but it seems like it might be of general
interest, so I'm reposting it. - MJD]

People I talk to often don't understand how authors get paid. It's
interesting, so I thought I'd send out a note about it.

Basically, the deal is that you get a percentage of the publisher's
net revenues. This percentage is called "royalties". So you're
getting a percentage of every book sold. Typical royalties seem to be
around 15%. O'Reilly's are usually closer to 10%. If there are
multiple authors, they split the royalty between them.

Every three or six months your publisher will send you a statement
that says how many copies sold and at what price, and what your
royalties are. If the publisher owes you money, the statement will be
accompanied by a check.

The 15% royalty is a percentage of the net receipts. The publisher
never sees a lot of the money you pay for the book in a store. Say
you buy a book for $60 in a bookstore. About half of that goes to the
store and the book distributor. The publisher gets the other half.
So the publisher has sold the book to the distributor for $30, and the
distributor sold it to the store for perhaps $45. This is why
companies like Amazon can offer such a large discount: there's no
store and no distributor.

So let's apply this information to a practical example and snoop into
someone else's finances. Perl Cookbook sells for $50.
Of that $50, O'Reilly probably sees about $25. Of that $25, about
$2.50 is authors' royalties. Assuming that Tom and Nat split the
royalties evenly (which perhaps they didn't; Tom was more important
than Nat) each of them gets about $1.25 per copy sold. Since O'Reilly
claims to have sold 150,000 copies of this book, we can guess that Tom
has made around $187,500 from this book. Maybe. It might be more (if
Tom got more than 50%) and it might be less (that 150,000 might
include foreign sales, for which the royalty might be different, or
bulk sales, for which the publisher might discount the cover price;
also, a lot of those 150,000 copies were the first edition, and I
forget the price of that.) But we can figure that Tom and Nat did
pretty well from this book. On the other hand, if $187,500 sounds
like a lot, recall that that's the total for 8 years, averaging about
$23,500 per year, and also recall that, as Nat says, writing a book
involves staring at the blank page until blood starts to ooze from
your pores.

Here's a more complicated example. The book Best of The Perl
Journal vol. 1 is a collection of articles by many people. The
deal these people were offered was that if they contributed less than
X amount, they would get a flat $150, and if they contributed
more than X amount, they would get royalties in proportion to
the number of pages they contributed. (I forget what X was.)
I was by far the contributor of the largest number of pages, about 14%
of the entire book. The book has a cover price of $40, so O'Reilly's
net revenues are about $20 per copy and the royalties are about $2 per
copy. Of that $2, I get about 14%, or $0.28 per copy. But for
Best of the Perl Journal, vol. 2, I contributed only one
article and got the flat $150. Which one was worth more for me? I
think it was probably volume 1, but it's closer than you might think.
There was a biggish check of a hundred and some dollars when the book
was first published, and then a smaller check, and by now the checks
are coming in amounts like $20.55 and $12.83.

The author only gets the 15% on the publisher's net receipts. If the
books in the stores aren't selling, the bookstore gets to return them
to the publisher for a credit. The publisher subtracts these copies
from the number of copies sold to arrive at the royalty. If more
copies come back than are sold, the author ends up owing the publisher
money! Sometimes when the book is a mass-market paperback, the
publisher doesn't want the returned copies; in this case the store is
supposed to destroy the books, tear off the covers, and send the
covers back to the publisher to prove that the copies didn't sell.
This saves on postage and trouble. Sometimes you see these coverless
books appear for sale anyway.

When you sign the contract to write the book, you usually get an
"advance". This is a chunk of money that the publisher pays you in
advance to help support you while you're writing. When you hear about
authors like Stephen King getting a one-million-dollar advance, this
is what they are talking about. But the advance is really a loan; you
pay it back out of your royalties, and until the advance is repaid,
you don't see any royalty checks. If you write the book and then it
doesn't sell, you don't get any royalties, but you still get to keep
the advance. But if you don't write the book, you usually have
to return the advance, or most of the advance. I've known authors who
declined to take an advance, but it seems to me that there is no
downside to getting as big an advance as possible. In the worst case,
the book doesn't sell, and then you have more money than you would
have gotten from the royalties. If the book does sell, you have the
same amount of money, but you have it sooner. I got a big advance for
HOP. My advance
will be paid back after 4,836 copies are sold. Exercise: estimate the
size of my advance. (Actually, the 4,836 is not quite correct,
because of variations in revenues from overseas sales, discounted
copies, and such like. When the publisher sells a copy of the book
from their web site, it costs the buyer $51 instead of $60, but the
publisher gets the whole $51, and pays royalties on the full amount.)

If the publisher manages to exploit the book in other ways, the author
gets various percentages. If Morgan Kaufmann produces a Chinese
translation of HOP, I get 5% of the revenues for each
copy; if instead they sell to a Chinese publisher the rights to
produce and sell a Chinese translation, I get 50% of whatever the
Chinese publisher paid them. If Universal pictures were to pay my
publisher a million dollars for the rights to make HOP
into a movie starring Kevin Bacon, I would get $50,000 of that.
(Wouldn't it be cool to live in that universe? I hear that 119 is a
prime number over there.)

[ Addendum 20060109: I was inspired to repost this by the arrival in
the mail today of my O'Reilly quarterly royalty statement. I thought
I'd share the numbers. Since the last statement, 31 copies of
Computer Science & Perl Programming were sold: 16
copies domestically and 15 foreign. The cover price is $39.95, so we
would expect that O'Reilly's revenues would be close to $619.22; in
fact, they reported revenues of $602.89. My royalty is 1.704 percent.
The statement was therefore accompanied by a check for $10.27. Who
says writing doesn't pay? ]

[ Addendum 20140428: The original source of Nat's remark about
writing is from Gene Fowler, who
said “Writing is easy. All you do is stare at a blank sheet of paper
until drops of blood form on your forehead.” ]

Earlier this winter I was reading Liber Abaci, which is
the book responsible for the popularity and widespread adoption of
Hindu-Arabic numerals in Europe. It was written in 1202 by Leonardo
of Pisa, who afterwards was called "Fibonacci".

Leonardo Pisano has an interesting notation for fractions. He often
uses mixed-base systems. In general, he writes:>

where a, b, c, p, q, r, x are integers. This represents the number:

which may seem odd at first. But the notation is a good one, and
here's why. Suppose your currency is pounds, and there are 20 soldi
in a pound and 12 denari in a soldo. This is the ancient Roman
system, used throughout Europe in the middle ages, and in England up
through the 1970s. You have 6 pounds, 7 soldi, and 4 denari. To
convert this to pounds requires no arithmetic whatsooever; the answer
is simply

And in general, L pounds, S soldi and D denari can be written

Now similarly you have a distance of 3 miles, 6 furlongs, 42 yards, 2
feet, and 7 inches. You want to calculate something having to do with
miles, so you need to know how many miles that is. No problem; it's
just

We tend to do this sort of thing either in decimals (which is
inconvenient unless all the units are powers of 10) or else we reduce
everything to a single denominator, in which case the numbers get
quite large. If you have many mixed units, as medieval merchants did,
Leonardo's notation is very convenient.

One operation that comes up all the time is as follows. Suppose you
have

and you wish that the denominator were s instead of b. It is easy to
convert. You calculate the quotient q and remainder r of dividing a·s
by b. Then the equivalent fraction with denominator s is just:

(Here we have replaced c + a/b with c + q/s + r/bs, which we can do
since q and r were chosen so that qb + r = as.)

Why would you want to convert a denominator in this way? Here is a
typical example. Suppose you have 24 pounds and you want to split it
11 ways. Well, clearly each share is worth

pounds; we can get that far without medieval arithmetic tricks. But
how much is 2/11 pounds? Now you want to convert the 2/11 to soldi;
there are 20 soldi in a pound. So you multiply 2/11 by 20 and get
40/11; the quotient is 3 and the remainder 7, so that each share is
really worth

pounds. That is, each share is worth 2 pounds, plus 3 and 7/11 soldi.

But maybe you want to convert the 7/11 soldi to denari; there are 12
denari in a soldo. So you multiply 7/11 by 12 and get 84/11; the
quotient is 7 and the remainder 7 also, so that each share is

Note that this system subsumes the usual decimal system, since you can
always write something like

when you mean the number 7.639. And in fact Leanardo does do
exactly this when it makes sense in problems concerning decimal
currencies and other decimal matters. For example, in Chapter 12 of
Liber Abaci, Leonardo says that there is a man with 100
bezants, who travels through 12 cities, giving away 1/10 of his money
in each city. (A bezant was a medieval coin minted in Constantinople.
Unlike the European money, it was a decimal currency, divided into 10
parts.) The problem is clearly to calculate (9/10)12 &times 100, and
Leonardo indeed gives the answer as

He then asks how much money was given away, subtracting the previous
compound fraction from 100, getting

(Leonardo's extensive discussion of this problem appears on
pp. 439–443 of L. E. Sigler Fibonacci's Liber Abaci: A Translation
into Modern English of Leonardo Pisano's Book of Calculation.)

The flexibility of the notation appears in many places. In a problem
"On a Soldier Receiving Three Hundred Bezants for His Fief" (p. 392)
Leonardo must consider the fraction

(That is, 1/534) which he multiplies by 2050601250 to get the final
answer in beautiful base-53 decimals:

Post scriptum: The Roman names for pound, soldo, and denaro are librum
("pound"), solidus, and denarius. The names survived into Renaissance
French (livre, sou, denier) and almost survived into modern English.
Until the currency was decimalized, British money amounts were written
in the form "£3 4s. 2d." even though the English names of the units
are "pound", "shilling", "penny", because "£" stands for "libra", "s"
for "solidi", and "d" for "denarii". In small amounts one would write
simply "10/6" instead of "10s. 6d.", and thus the "/" symbol came to
be known as a "solidus", and still is.

The Spanish word for money, dinero, means denarii. The
Arabic word dinar, which is the name of the currency in
Algeria, Bahrain, Iraq, Jordan, Kuwait, Libya, Sudan, and Tunisia,
means denarii.

The Principal Navigations, Voyages, Traffiques, & Discoveries of
the English Nation was published in 1589, a collection of
essays, letters, and journals written mostly by English persons about
their experiences in the great sea voyages of discovery of the latter
half of the 16th century.

One important concern of the English at this time was to find an
alternate route to Asia and the spice islands, since the Portuguese
monopolized the sea route around the coast of Africa. So many of the
selections concern the search for a Northwest or Northeast passage,
routes around North America or Siberia, respectively. Other items
concern military battles, including the defeat of the Spanish Armada;
proper outfitting of whaling ships, and an account of a sailor who was
shipwrecked in the West Indies and made his way home at last sixteen
years later.

One item, titled "Experiences and reasons of the Sphere, to proove all
partes of the worlde habitable, and thereby to confute the position of
the five Zones," contains the following sentence:

First you are to understand that the Sunne doeth worke his
more or lesse heat in these lower parts by two meanes, the one
is by the kinde of Angle that the Sunne beames doe make with
the earth, as in all Torrida Zona it maketh perpendicularly
right Angles in some place or other at noone, and towards the
two Poles very oblique and uneven Angles.

This explanation is quite correct. (The second explanation, which I
omitted, is that the sun might spend more or less time above the
horizon, and is also correct.) This was the point at which I happened
to set down the book before I went to sleep.

But over the next couple of days I realized that there was something
deeply puzzling about it: This explanation should not be accessible to
an Englishman of 1580, when this item was written.

In 2006, I would explain that the sun's rays are a directed radiant
energy field in direction E, and that the energy received by a
surface S is the dot product of the energy vector E and
the surface normal vector n. If E and n are
parallel, you get the greatest amount of energy; as E and
n become perpendicular, less and less energy is incident on
S.

Englishmen in 1580 do not have a notion of the dot product of two
vectors, or of vectors themselves for that matter. Analytic geometry
will not be invented until 1637. You can explain the weakness of
oblique rays without invoking vectors, by appeal to the law of
conservation of energy, but the Englishmen do not have the idea of an
energy field, or of conservation of energy, or of energy at all. They
do not have any notion of the amount of radiant energy per unit area.
Galileo's idea of mathematical expression of physical law will not be
published until 1638.

So how do they have the idea that perpendicular sun is more intense
than oblique? How did they think this up? And what is their argument
for it?

(Try to guess before you read the answer.)

In fact, the author is completely wrong about the reason. Here's what
he says the answer is:

... the perpendicular beames reflect and reverberate in
themselves, so that the heat is doubled, every beam striking
twice, & by uniting are multiplied, and continued strong in
forme of a Columne. But in our Latitude of 50. and
60. degrees, the Sunne beames descend oblique and slanting
wise, and so strike but once and depart, and therefore our
heat is the lesse for any effect that the Angle of the Sunne
beames make.

Did you get that? Perpendicular sun is warmer because the beams get
you twice, once on the way down and once on the way back up. But
oblique beams "strike but once and depart."

One of the types of problems that al-Khwarizmi treats extensively in
his book is the problem of computing inheritances under Islamic law.

Well, these examples are getting sillier and sillier, but

... but maybe they're not silly enough:

A rope over the top of a fence has the same length on each
side and weighs one-third of a pound per foot. On one end of
the rope hangs a monkey holding a banana, and on the other end
a weight equal to the weight of the monkey. The banana weighs
2 ounces per inch. The length of the rope in feet is the same
as the age of the monkey, and the weight of the monkey in
ounces is as much as the age of the monkey's mother. The
combined ages of the monkey and its mother is 30
years. One-half the weight of the monkey plus the weight of
the banana is one-fourth the sum of the weights of the rope
and the weight. The monkey's mother is one-half as old as the
monkey will be when it is three times as old as its mother was
when she was one-half as old as the monkey will be when it is
as old as its mother will be when she is four times as old as
the monkey was when it was twice as old as its mother was when
she was one-third as old as the monkey was when it was as old
as its mother was when she was three times as old as the
monkey was when it was one-fourth as old as its is now. How
long is the banana?

Islamic inheritance law
Since Linogram
incorporates a generic linear equation solver, it can be used to solve
problems that have nothing to do with geometry. I don't claim that
the following problem is a "practical application" of
Linogram or of the techniques of HOP. But I did find it
interesting.

An early and extremely influential book on algebra was Kitab
al-jabr wa l-muqabala ("Book of Restoring and Balancing"),
written around 850 by Muhammad ibn Musa al-Khwarizmi. In fact, this
book was so influential in Europe that it gave us the word "algebra",
from the "al-jabr" in the title. The word "algorithm" itself is a
corrupted version of "al-Khwarizmi".

One of the types of problems that al-Khwarizmi treats extensively in
his book is the problem of computing inheritances under Islamic law.
Here's a simple example:

A woman dies, leaving her husband, a son, and three daughters.

Under the law of the time, the husband is entitled to 1/4 of the
woman's estate, daughters to equal shares of the remainder, and sons
to shares that are twice the daughters'. So a little arithmetic will
find that the son receives 30%, and the three daughters 15% each. No
algebra is required for this simple problem.

Here's another simple example, one I made up:

A man dies, leaving two sons. His estate consists of ten
dirhams of ready cash and ten dirhams as a claim against one
of the sons, to whom he has loaned the money.

There is more than one way to look at this, depending on the
prevailing law and customs. For example, you might say that the death
cancels the debt, and the two sons each get half of the cash, or 5
dirhams. But this is not the solution chosen by al-Khwarismi's
society. (Nor by ours.)

Instead, the estate is considered to have 20 dirhams in assets, which
are split evenly between the sons. Son #1 gets half, or 10 dirhams,
in cash; son #2, who owes the debt, gets 10 dirhams, which cancels his
debt and leaves him at zero.

This is the fairest method of allocation because it is the only one that
gives the debtor son neither an incentive nor a disincentive to pay
back the money. He won't be withholding cash from his dying father in
hopes that dad will die and leave him free; nor will he be out
borrowing from his friends in a desparate last-minute push to pay back
the debt before dad kicks the bucket.

However, here is a more complicated example:

A man dies, leaving two sons and bequeathing one-third of his
estate to a stranger. His estate consists of ten dirhams of
ready cash and ten dirhams as a claim against one of the sons,
to whom he has loaned the money.

According to the Islamic law of the time, if son #2's share of the
estate is not large enough to enable him to pay back the debt
completely, the remainder is written off as uncollectable. But the
stranger gets to collect his legacy before the shares of the rest of
the estate is computed.

We need to know son #2's share to calculate the writeoff. But we need
to know the writeoff to calculate the value of the estate, we need the
value of the estate to calculate the bequest to the stranger, and the
need to know the size of the bequest to calculate the size of the
shares. So this is a problem in algebra.

In Linogram, we write:

number estate, writeoff, share, stranger, cash = 10, debt = 10;

Then the laws introduce the following constraints: The total estate is
the cash on hand, plus the collectible part of the debt:

constraints {
estate = cash + (debt - writeoff);

Son #2 pays back part of his debt with his share of the estate; the
rest of the debt is written off:

debt - share = writeoff;

The stranger gets 1/3 of the total estate:

stranger = estate / 3;

Each son gets a share consisting of half of the value of the estate
after the stranger is paid off:

share = (estate - stranger) / 2;
}

Linogram will solve the equations and try to "draw" the result. We'll
tell Linogram that the way to "draw" this "picture" is just to print
out the solutions of the equations:

Here's the solution found by Linogram: Of the 10 dirham debt owed by
son #2, 5 dirhams are written off. The estate then consists of 10
dirhams in cash and the 5 dirham debt, totalling 15. The stranger
gets one-third, or 5. The two sons each get half of the remainder, or
5 each. That means that the rest of the cash, 5 dirhams, goes to son
#1, and son #2's inheritance is to have the other half of his debt
cancelled.

OK, that was a simple example. Suppose the stranger was only supposed
to get 1/4 of the estate instead of 1/3? Then the algebra works out
differently:

Here only 4 dirhams of son #2's debt is forgiven. This leaves an
estate of 16 dirhams. 1/4 of the estate, or 4 dirhams, is given to
the stranger, leaving 12. Son #1 receives 6 dirhams cash; son #2's 6
dirham debt is cancelled.

(Notice that the rules always cancel the debt; they never leave anyone
owing money to either their family or to strangers after the original
creditor is dead.)

These examples are both quite simple, and you can solve them in your
head with a little tinkering, once you understand the rules. But here
is one that al-Khwarizmi treats that I could not do in my head; this
is the one that inspired me to pull out Linogram. It is as before,
but this time the bequest to the stranger is 1/5 of the estate plus
one extra dirham. The Linogram specification is:

One of the types of problems that al-Khwarizmi treats extensively in
his book is the problem of computing inheritances under Islamic law.

Well, these examples are getting sillier and sillier, but I still think
they're kind of fun. I was thinking about the Islamic estate law stuff,
and I remembered that there's a famous problem, perhaps invented about
1500 years ago, that says of the Greek mathematician Diophantus:

He was a boy for one-sixth of his life. After one-twelfth more, he
acquired a beard. After another one-seventh, he married. In the
fifth year after his marriage his son was born. The son lived half
as many as his father. Diophantus died 4 years after his son. How
old was Diophantus when he died?

84 years is indeed the correct solution. (That is a really annoying
roundoff error. Gosh, how I hate floating-point numbers! The next
version of Linogram is going to use rationals everywhere.)

Note that because all the variables except for "life" are absolute dates
rather than durations, they are all indeterminate. If we add the
equation "birth = 211" for example, then we get an output that lists the
years in which all the other events occurred.